F?.  S.  HuTCHEON 


LIBRARY 

THE  UNIVERSITY 
OF  CALIFORNIA 

SANTA  BARBARA 

PRESENTED  BY 
ELEANOR  HUTCHEON 


WORKS  OF 
THE  LATE   JOSEPH  LIPKA 

PUBLISHED    BY 

JOHN  WILEY  &  SONS,  INC. 


Graphical  and  Mechanical  Computation 

An  aid  in  the  solution  of  a  large  number  of  problems 
which  the  engineer,  as  well  as  the  student  of  engineering, 
meets  in  his  work,  ix  +  264  pages.  6  by  9.  207  fig- 
ures, 2  charts.  Cloth. 

Also  published  in  two  parts 

Part  I.  Alignment  Charts.  xiv  +  119  pages.  6  by  9. 
130  figures,  2  charts.  Cloth. 

Part  II.  Experimental  Data.  Pages  120  to  259.  6  by  9. 
77  figures.  Cloth. 

BY  S.  R.  CUMMINGS  AND  JOSEPH  LIPKA 

Alignment  Charts  for  the  Engineer 

By  S.  R.  Cummings.S.M.,  Research  Engineer,  The  Hoo- 
ver Co.,  and  the  late  Joseph  Lipka,  Ph.  D. 
Part  I.  Air  and  Steam.  Twenty  charts  for  various  en- 
gineering equations  and  formulas,  designed  for  practical 
use  by  the  engineer  and  student  of  engineering.  9$  by 
12.  Loose  leaf,  in  heavy  paper  envelope. 

BY  HUDSON,  LIPKA,  LUTHER  AND  PEABODY 

The  Engineers'  Manual 

By  Ralph  G.  Hudson,  S.  B.,  Professor  of  Electrical 
Engineering,  Massachusetts  Institute  of  Technology, 
assisted  by  the  late  Joseph  Lipka,  Ph.D.,  Howard  B. 
Luther,  S.  B.,  Dipl.  Ing.,  Professor  of  Civil  Engineer- 
ing, University  of  Cincinnati,  and  Dean  Peabody,  Jr.,  S. 
B.,  Associate  Professor  of  Applied  Mechanics,  Massachu- 
setts Institute  of  Technology. 

A  consolidation  of  the  more  commonly  used  formulas  of 
engineering,  each  arranged  with  a  statement  of  its  appli- 
cation. Second  edition,  iv  +  340  pages.  5  by  7|.  238 
figures.  Flexible  binding. 

BY  R.  G.  HUDSON  AND  JOSEPH  LIPKA 

A  Manual  of  Mathematics 

By  Ralph  G.  Hudson,  S.B.,  and  the  late  Joseph  Lipka, 
Ph.  D. 

A  collection  of  mathematical  tables  and  formulas  cover- 
ing the  subjects  most  generally  used  by  engineers  and 
by  students  of  mathematics,  and  arranged  for  quick  ref- 
erence, iii  -(-  132  pages.  5  by  7J.  95  figures.  Flexible 
binding. 

A  Table  of  Integrals 

By  Ralph  G.  Hudson,  S.B.,  and  the  late  Joseph  Lipka, 
Ph.D. 

Contains  a  Table  of  Derivatives,  Table  of  Integrals,  Nat- 
ural Logarithms,  Trigonometric  and  Hyperbolic  Func- 
tions. 24  pages.  5  by  7J.  Paper. 


[GRAPHICAL  AND  MECHANICAL 
COMPUTATION/ 

PART  n.    EXPERIMENTAL  DATA 


BY 

JOSEPH  LIPKA,  PH.D. 

LATE  ASSISTANT  PROFESSOR  OF  MATHEMATICS  IN  THE  MASSACHUSETTS 
INSTITUTE  OF  TECHNOLOGY 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:   CHAPMAN   &   HALL,   LIMITED 


«** 


IN  THE  REPRINTING  OF  THIS  BOOK,  THE  RECOM- 
MENDATIONS OF  THE  WAR  PRODUCTION  BOARD 
HAVE  BEEN  OBSERVED  FOR  THE  CONSERVATION 
OF  PAPER  AND  OTHER  IMPORTANT  WAR  MA- 
TERIALS. THE  CONTENT  REMAINS  COMPLETE 
AND  UNABRIDGED. 


COPYRIGHT,  1918, 

BY 

JOSEPH  LIPKA 


Printed  in  U.  S.  A. 


PREFACE 


This  book  embodies  a  course  given  by  the  writer  for  a  number  of 
years  in  the  Mathematical  Laboratory  of  the  Massachusetts  Institute 
of  Technology.  It  is  designed  as  an  aid  in  the  solution  of  a  large  num- 
ber of  problems  which  the  engineer,  as  well  as  the  student  of  engineering, 
meets  in  his  work. 

In  the  opening  chapter,  the  construction  of  scales  naturally  leads  to 
a  discussion  of  the  principles  upon  which  the  construction  of  various 
slide  rules  is  based.  The  second  chapter  develops  the  principles  of  a 
network  of  scales,  showing  their  application  to  the  use  of  various  kinds 
of  coordinate  paper  and  to  the  charting  of  equations  in  three  variables. 

Engineers  have  recognized  for  a  long  time  the  value  of  graphical 
charts  in  lessening  the  labor  of  computation.  Among  the  charts  devised 
none  are  so  rapidly  constructed  nor  so  easily  read  as  the  charts  of  the 
alignment  or  nomographic  type  —  a  type  which  has  been  most  fully 
developed  by  Professor  M.  d'Ocagne  of  Paris.  Chapters  III,  IV,  and  V 
aim  to  give  a  systematic  development  of  the  construction  of  alignment 
charts;  the  methods  are  fully  illustrated  by  charts  for  a  large  number 
of  well-known  engineering  formulas.  It  is  the  writer's  hope  that  the 
simple  mathematical  treatment  employed  in  these  chapters  will  serve  to 
make  the  engineering  profession  more  widely  acquainted  with  this  time 
and  labor  saving  device. 

Many  formulas  in  the  engineering  sciences  are  empirical,  and  the 
value  of  many  scientific  and  technical  investigations  is  enhanced  by  the 
discovery  of  the  laws  connecting  the  results.  Chapter  VI  is  concerned 
with  the  fitting  of  equations  to  empirical  data.  Chapter  VII  considers 
the  case  where  the  data  are  periodic,  as  in  alternating  currents  and  volt- 
ages, sound  waves,  etc.,  and  gives  numerical,  graphical,  and  mechanical 
methods  for  determining  the  constants  in  the  equation. 

When  empirical  formulas  cannot  be  fitted  to  the  experimental  data, 
these  data  may  still  be  efficiently  handled  for  purposes  of  further 
computation,  —  interpolation,  differentiation,  and  integration,  —  by  the 
numerical,  graphical,  and  mechanical  methods  developed  in  the  last 
two  chapters. 

Numerous  illustrative  examples  are  worked  throughout  the  text, 
and  a  large  number  of  exercises  for  the  student  is  given  at  the  end  of 
each  chapter.  The  additional  charts  at  the  back  of  the  book  will  serve 

iii 


iv  PREFACE 

as 'an  aid  in  the  construction  of  alignment  charts.     Bibliographical 
references  will  be  found  in  the  footnotes. 

The  writer  wishes  to  express  his  indebtedness  for  valuable  data  to 
the  members  of  the  engineering  departments  of  the  Massachusetts 
Institute  of  Technology,  and  to  various  mathematical  and  engineering 
publications.  He  owes  the  idea  of  a  Mathematical  Laboratory  to 
Professor  E.  T.  Whittaker  of  the  University  of  Edinburgh.  He  is 
especially  indebted  to  Capt.  H.  M.  Brayton,  U.  S.  A.,  a  former  student, 
for  his  valuable  suggestions  and  for  his  untiring  efforts  in  designing  a 
large  number  of  the  alignment  charts.  Above  all  he  is  most  grateful  to 
his  wife  for  her  assistance  in  the  revision  of  the  manuscript  and  the 
reading  of  the  proof,  and  for  her  constant  encouragement  which  has 
greatly  lightened  the  labor  of  writing  the  book. 

JOSEPH   LIPKA. 

CAMBRIDGE,  MASS., 
Oct.  13,  1918. 


CONTENTS. 

CHAPTER   I. 

SCALES  AND   THE   SLIDE  RULE. 
ART.  PAGE 

1.  Definition  of  a  scale I 

2.  Representation  of  a  function  by  a  scale I 

3.  Variation  of  the  scale  modulus 2 

4.  Stationary  scales 5 

5.  Sliding  scales ' 7 

6.  The  logarithmic  slide  rule 9 

7.  The  solution  of  algebraic  equations  on  the  logarithmic  slide  rule 1 1 

8.  The  log-log  slide  rule . 13 

9.  Various  other  straight  slide  rules 15 

10.  Curved  slide  rules 16 

Exercises 18 

CHAPTER   II. 

NETWORK  OF  SCALES.     CHARTS  FOR  EQUATIONS  IN  TWO   AND 
THREE  VARIABLES. 

11.  Representation  of  a  relation  between  two  variables  by  means  of  perpen- 

dicular scales 20 

12.  Some  illustrations  of  perpendicular  scales 21 

13.  Logarithmic  coordinate  paper 22 

14.  Semilogarithmic  coordinate  paper 24 

15.  Rectangular  coordinate  paper  —  the  solution  of  algebraic  equations  of  the 

2nd,  3rd,  and  4th  degrees 26 

1 6.  Representation  of  a  relation  between  three  variables  by  means  of  perpen- 

dicular scales • 28 

17.  Charts  for  multiplication  and  division 30 

18.  Three-variable  charts.     Representing  curves  are  straight  lines 32 

19.  Rectangular  chart  for  the  solution  of  cubic  equations 35 

20.  Three- variable  charts.     Representing  curves  are  not  straight  lines 37 

21.  Use  of  three  indices.     Hexagonal  charts 40 

Exercises 42 

CHAPTER  III. 
NOMOGRAPHIC   OR  ALIGNMENT   CHARTS. 

22.  Fundamental  principle 44 

(I)  Equation  of  form  f^u)  +  f2(v)  =/3(w)    or  /,(«)  •  }M  =  MW)  —  Three 

parallel  scales 45~54 

23.  Chart  for  equation  (I) 45 

24.  Chart  for  multiplication  and  division 47 


vi  CONTENTS 

ART.  PAGE 

25.  Combination  chart  for  various  formulas 48 

26.  Grasshoff 's  formula  for  the  weight  of  dry  saturated  steam 50 

27.  Tension  in  belts  and  horsepower  of  belting 52 

(II)  Equation  of  form 

/i(«)  +/»(»)  +/•(»)  +•  •  •   =  /«(*)    or   /,(«)  ./,  (t>)  ./.(w)  .  .  .    =  /<(*) 
—  Four  or  more  parallel  scales 55~63 

28.  Chart  for  equation  (II) 55 

29.  Chezy  formula  for  the  velocity  of  flow  of  water  in  open  channels 56 

30.  Hazen-Williams  formula  for  the  velocity  of  flow  of  water  in  pipes 57 

31.  Indicated  horsepower  of  a  steam  engine 6l 

Exercises , 64 


CHAPTER   IV. 
NOMOGRAPHIC   OR  ALIGNMENT   CHARTS    (Continued). 

(HI)  Equation  of  form/i(«)  =  /2(»)  •  MW)  or/i(«)  =  ft(v)f*w  —  Z  chart. .     65-67 

32.  Chart  for  equation  (III) 65 

33.  Tension  on  bolts  with  U.  S.  standard  threads 66 

(IV)  Equation  of  form    '        =   *       — Two  intersecting  index  lines 68-75 

34.  Chart  for  equation  (IV) 68 

35.  Prony  brake  or  electric  dynamometer  formula 69 

36.  Deflection  of  beam  fixed  at  ends  and  loaded  at  center 70 

37.  Deflection  of  beams  under  various  methods  of  loading  and  supporting 71 

38.  Specific  speed  of  turbine  and  water  wheel 73 

(V)  Equation  of  form  fi(u)  =  fa(v)  '  fz(w)  •/•(/)  .  .  .  —  Two  or  more  inter- 

*          secting  index  lines '. 76-87 

39.  Chart  for  equation  (V) 76 

40.  Twisting  moment  in  a  cylindrical  shaft 77 

41.  D'Arcy's  formula  for  the  flow  of  steam  in  pipes 79 

42.  Distributed  load  on  a  wooden  beam 80 

43.  Combination  chart  for  six  beam  deflection  formulas 84 

44.  General  considerations 87 

(VI)  Equation  of  form  -JT\  =  fT\  —  Parallel  or  perpendicular  index  lines    87-91 

45.  Chart  for  equation  (VI) 87 

46.  Weight  of  gas  flowing  through  an  orifice 89 

47.  Armature  or  field  winding  from  tests 90 

48.  Lame  formula  for  thick  hollow  cylinders  subjected  to  internal  pressure 91 

(VII)  Equation      of     form     /,(«)  -  /,(»)  =  /3(w)  -  ft(q)      or     /i(«)  :/«(*) 

=  /a(ttO  :  /»(?)  —  Parallel  or  perpendicular  index  lines 9i~95 

49.  Chart  for  equation  (VII) 91 

50.  Friction  loss  in  flow  of  water 94 

Exercises 95 


CONTENTS  YD 

CHAPTER  V. 
HOMOGRAPHIC  OR  ALIGHMEHT  CHARTS  (Continued). 


(VIE)  Equation  of  form  /,(«)  +/,(*)  =  —  Parallel  or  perpendicular 

index  lines  ...................................................  97-104 

51.  Chart  for  equation  (Vlli)  ...........................................  97 

52.  Moment  of  inertia  of  cylinder  ........................................  99 

53.  Bazin  formula  for  velocity  of  flow  in  open  channels  .....................  101 

54.  Resistance  of  riveted  steel  plate  .....................................  ".  IOT 

(IX)  Equation  of  form 


Three  or  more  concurrent  scales 104—106 

55.  Chart  for  equation  (IX) 104 

56.  Focal  length  of  a  lens 106 

(X)  Equation  of  form  /.(»)+/,(*)-/,(«)  =/«(»)  — Straight   and  curved 
scales 106-113 

57.  Chart  for  equation  (X) 106 

58.  Storm  water  run-off  formula 107 

59.  Francis  formula  for  a  contracted  weir no 

60.  The  solution  of  cubic  and  quadratic  equations no 

(XI)  Additional  forms  of  equations.     Combined  methods 114-117 


62.  Chart  for  equation  of  form  /,(«)+/,(*)  •/,(*>)  =/«(f)  ...................     114 

63.  Chart  for  equation  of  form  /,(«)  ./4(a)  +/^»)  •/,(»)  =  1  .................     114 


64.  Cfc«tforeo^ionoffann+=...  .....  05 

65.  Chartforequadonofform^)+^  =  i  ......................  .....  .„ 

66.  Chart  for  equation  of  form/if.)  -/.(j)  +/,(*)  -fifa)  =  /5(«.)  ..............  116 

67.  Chart  for  equation  of  form  /!(«)•/,(«)+/,(»)  •/4(w)=/i(J)+/«(w)  .......  117 

............................................  "7 

for  Chapters  III,  IV,  V  .........................  118 


CHAPTER  YL 
EMPIRICAL  FORMULAS  — If OH-PKRIODIC  CURVES. 

68.  Experimental  data lao 

(C  The  straight  line 122-127 

69.  The  straight  fine,  j  =  Joe 122 

70.  The  straight  Foe,  jr  =  «  +  fcr 125 


V1H  CONTENTS 

ART.  PAGE 

(II)  Formulas  involving  two  constants 128-139 

71.  Simple  parabolic  and  hyperbolic  curves,  y  =  a*6 128 

72.  Simple  exponential  curves,  y  =  a^z 131 

73.  Parabolic  or  hyperbolic  curve,  y  =  a  +  bxn  (wnere  n  is  Known) 135 

74.  Hyperbolic  curve,  y  =          ,    ,  or  -  =  a  +  bx 137 

(III)  Formulas  involving  three  constants 140-152 

75.  The  parabolic  or  hyperbolic  curve,  y  =  axb  +  c 140 

76.  The  exponential  curve,  y  =  aebx  +  c 142 

77.  The  parabola,  y  =  a  +  bx  +  ex2 145 

78.  The  hyperbola,  y  =  — \-  c 149 


79-   The  logarithmic  or  exponential  curve,  log  y  =  a  +  bx  -\-  ex2  or  y  =  aePI~*~cx  .  .  151 

(IV)  Equations  involving  four  or  more  constants 152-164 

80.  The  additional  terms  cedx  and  cxd 152 

81.  The  equation  y  =  a  +  bx  +  cedx 153 

82.  The  equation  y  =  aebx  +  cedx 156 

83.  The  polynomial  y  =  a  +  bx  +  ex2  +  dx3  + 159 

84.  Two  or  more  equations 161 

Exercises 164 

CHAPTER  VII. 
EMPIRICAL  FORMULAS  —  PERIODIC   CURVES. 

85.  Representation  of  periodic  phenomena 170 

86.  The  fundamental  and  the  harmonics  of  a  trigonometric  series 170 

87.  Determination  of  the  constants  when  the  function  is  known 173 

88.  Determination  of  the  constants  when  the  function  is  unknown 174 

89.  Numerical  evaluation  of  the  coefficients.     Even  and  odd  harmonics 179 

90.  Numerical  evaluation  of  the  coefficients.     Odd  harmonics  only 186 

91.  Numerical  evaluation  of  the  coefficients.     Averaging  selected  ordinates.  ...  .192 

92.  Numerical   evaluation   of  the  coefficients.     Averaging  selected   ordinates. 

Odd  harmonics  only 198 

93.  Graphical  evaluation  of  the  coefficients 200 

94.  Mechanical  evaluation  of  the  coefficients.     Harmonic  analyzers 203 

Exercises 207 

CHAPTER  VIII. 
INTERPOLATION. 

95.  Graphical  interpolation 209 

96.  Successive  differences  and  the  construction  of  tables  210 

97.  Newton's  interpolation  formula 214 

98.  Lagrange's  formula  of  interpolation 218 

99.  Inverse  interpolation 2IO 

Exercises 22  ,£ 


CONTENTS 


CHAPTER   IX. 

APPROXIMATE  INTEGRATION  AND   DIFFERENTIATION. 

ART.  PAGE 

100.  The  necessity  for  approximate  methods 224 

101.  Rectangular,  trapezoidal,  Simpson's,  and  Durand's  rules 224 

102.  Applications  of  approximate  rules 227 

103.  General  formula  for  approximate  integration 231 

104.  Numerical  differentiation 234 

105.  Graphical  integration 237 

106.  Graphical  differentiation 244 

107.  Mechanical  integration.     The  planimeter 246 

108.  Integrators 250 

109.  The  integraph 252 

no.    Mechanical  differentiation.     The  differentiator 255 

Exercises 256 


CHAPTER  VI. 
EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES. 

68.  Experimental  data.  —  In  scientific  or  technical  investigations  we 
are  often  concerned  with  the  observation  or  measurement  of  two  quanti- 
ties, such  as  the  distance  and  the  time  for  a  freely  falling  body,  the  volume 
of  carbon  dioxide  dissolving  in  water  and  the  temperature  of  the  water, 
the  load  and  the  elongation  of  a  certain  wire,  the  voltage  and  the  current 
of  a  magnetite  arc,  etc.  The  results  of  a  series  of  measurements  of  the 
same  two  quantities  under  similar  conditions  are  usually  presented  in  the 
form  of  a  table.  Thus  the  following  table  gives  the  results  of  observa- 
tions on  the  pressure  p  of  saturated  steam  in  pounds  per  sq.  in.  and  the 
volume  v  in  cu.  ft.  per  pound : 

p  =       10  20  30  40  50  60 

v=      37-80         19.72         13.48         10.29  8.34  6.62 


40 
30 
20 
10 

0 
I 

\ 

\ 

\ 

k 

\ 

^ 

'•—  . 

--^^ 

•**-^ 

)            10           20          30          40           50          60 
(P) 

FIG.  68. 

We  represent  these  results  graphically  by  plotting  on  coordinate  paper 
the  points  whose  coordinates  are  the  corresponding  values  of  the  measured 
quantities  and  by  drawing  a  smooth  curve  through  or  very  near  these 
points.  Fig.  68  gives  a  graphical  representation  of  the  above  table, 
where  the  values  of  p  are  laid  off  as  abscissas  and  the  values  of  v  as  ordi- 
nates  and  a  smooth  curve  is  drawn  so  as  to  pass  through  or  very  near  the 
plotted  points. 


ART.  68  EXPERIMENTAL  DATA  121 

The  fact  that  a  smooth  curve  can  be  drawn  so  as  to  pass  very  near  the 
plotted  points  leads  us  to  suspect  that  some  relation  may  exist  between 
the  measured  quantities,  which  may  be  represented  mathematically  by 
the  equation  of  the  curve.  Since  the  original  measurements,  the  plotting 
of  the  points,  and  the  drawing  of  the  curve  all  involve  approximations, 
the  equation  will  represent  the  true  relation  between  the  quantities  only 
approximately.  Such  an  equation  or  formula  is  known  as  an  empirical 
formula,  to  distinguish  it  from  the  equation  or  formula  which  expresses 
a  physical,  chemical,  or  biological  law.  A  large  number  of  the  formulas 
in  the  engineering  sciences  are  empirical  formulas.  Such  empirical  for- 
mulas may  then  be  used  for  the  purpose  of  interpolation,  i.e.,  for  comput- 
ing the  value  of  one  of  the  quantities  when  the  value  of  the  other  is  given 
within  the  range  of  values  used  in  determining  the  formula. 

It  is  at  once  evident  that  any  number  of  curves  can  be  drawn  so  as  to 
pass  very  near  the  plotted  points,  and  therefore  that  any  number  of 
equations  might  approximate  the  data  equally  well.  The  nature  of  the 
experiment  may  give  us  a  hint  as  to  the  form  of  the  equation  which  will 
best  represent  the  data.  Otherwise  the  problem  is  more  indeterminate. 
If  the  points  appear  to  lie  on  or  near  a  straight  line,  we  may  assume  an 
equation  of  the  first  degree,  y  =  a  +  bx,  in  the  variables.  But  if  the 
points  deviate  systematically  from  a  straight  line,  the  choice  of  an  equa- 
tion is  more  difficult.  Often  the  form  of  the  curve  will  suggest  the  type 
of  equation,  parabolic,  exponential,  trigonometric,  etc.,  but  in  all  cases, 
we  should  choose  an  equation  of  as  simple  a  form  as  possible.  Before 
proceeding  any  further  with  this  choice  we  may  test  the  correctness  of 
the  form  of  the  equation  by  "rectifying"  the  curve,  i.e.,  by  writing  the 
assumed  equation  in  the  form 

(i)  f(y}  =  a  +  bF(x)        or         (2)  /  =  a  +  bx', 

where  /  =  f(y)  and  x'  =  F(x),  and  plotting  the  points  with  x'  and  y' 
as  coordinates ;  if  the  points  of  this  plot  appear  to  lie  on  or  very  near  a 
straight  line,  then  this  line  can  be  represented  by  equation  (2)  and  hence 
the  original  curve  by  equation  (i).  We  shall  use  the  method  of  rectifica- 
tion quite  freely  in  the  work  which  follows. 

Having  chosen  a  simple  form  for  the  approximate  equation  we  now 
proceed  to  determine  the  approximate  values  of  the  constants  or  co- 
efficients appearing  in  the  equation.  The  method  of  approximation 
employed  in  determining  these  constants  depends  upon  the  desired  degree 
of  accuracy.  We  may  employ  one  of  three  methods:  the  method  of 
selected  points,  the  method  of  averages,  or  the  method  of  Least  Squares.  Of 
these,  the  first  is  the  simplest  and  the  approximation  is  close  enough 
for  a  large  number  of  problems  arising  in  technical  work;  the  second  re- 
quires a  little  more  computation  but  usually  gives  closer  approximations; 


122 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


while  the  third  gives  the  best  approximate  values  of  the  constants  but  the 
work  of  determining  these  values  is  quite  laborious.  All  three  methods 
will  be  illustrated  in  some  of  the  problems  which  follow. 

After  the  constants  have  been  determined  the  formula  should  be 
tested  by  performing  several  additional  experiments  where  the  variables 
lie  within  the  range  of  the^  previous  data,  and  comparing  these  results 
with  those  given  by  the  empirical  formula. 

We  shall  now  work  two  illustrative  examples  to  indicate  the  general 
method  of  procedure. 


(I)   THE   STRAIGHT  LINE. 

69.  The  straight  line,  y  —  bx.  —  The  following  table  gives  the  results 
of  a  series  of  experiments  on  the  determination  of  the  elongation  E  in 
inches  of  annealed  high  carbon  steel  wire  of  diameter  0.0693  in.  and  gage 
length  30  in.  due  to  the  load  W  in  pounds. 


w 

E 

EW 

W* 

Ec' 

£0" 

Ef 

A' 

A" 

A'" 

o 

O 

O 

0 

0 

0 

0 

O 

O 

O 

5° 

0.0130 

0.650 

2,500 

0.0130 

0.0131 

0.0131 

O 

—    I 

—    I 

100 

0.0251 

2.510 

10,000 

0.0260 

0.0261 

0.0262 

-  9 

—  10 

—  II 

15° 

0.0387 

5  805 

22,500 

0.0390 

0.0392 

0.0393 

-  3 

—  5 

-  6 

200 

0.0520 

10.400 

40,000 

0.0520 

0.0522 

0.0524 

o 

+    2 

—  4 

225. 

0.0589 

13-253 

50,625 

0.0585 

0.0587 

0.0589 

+  4 

+    2 

0 

250 

0.0659 

!6-475 

62,500 

0.0650 

0.0653 

0.0655 

+  9 

+  6 

+  4 

260 

0.0689 

!7-9!4 

67,600 

0.0676 

0.0679 

0.0681 

+13 

+  10 

+  8 

2  1235 

0.3^25 

67.007 

255-725 

38 

36 

34 

1-7-8=  4.8 

4-5 

4-3 

SA*  =  356 

270 

254 

The  plot.  —  The  data  are  plotted  on  a  sheet  of  coordinate  paper  about 
10  inches  square  and  ruled  in  twentieths  of  an  inch  or  in  millimeters.  If 
we  wish  to  express  the  elongation  as  a  function  of  the  load,  we  plot  the 
load  on  the  horizontal  axis  or  as  abscissas,  if  the  load  as  a  function  of  the 
elongation  we  plot  the  latter  as  abscissas.  In  Fig.  69  we  have  plotted 
the  values  of  W  as  abscissas  and  the  values  of  E  as  ordinates.  The  scales 
with  which  these  values  are  plotted  are  generally  chosen  so  that  the 
length  of  the  axis  represents  the  total  range  of  the  corresponding  vari- 
able, and  so  that  the  line  or  curve  is  about  equally  inclined  to  the  two 
axes.  There  is  no  advantage  in  choosing  the  scale  units  on  the  two  axes 
equal.  Care  should  be  taken  not  to  choose  the  units  either  too  small  or 
too  large;  for  in  the  former  case  the  precision  of  the  data  will  not  be 
utilized,  and  in  the  latter  case  the  deviations  from  a  representative  line 


ART.  69 


THE  STRAIGHT  LINE,  y  =  bx 


123 


or  curve  are  likely  to  be  magnified.  The  drawing  of  a  good  plot  is  evi- 
dently a  matter  of  judgment.  It  is  best  to  mark  the  plotted  points  as 
the  intersection  of  two  short  straight  lines,  one  horizontal  and  one 
vertical. 


0.07 
0.06 
0.05 

0.04 
(E) 
0,03 

0.02 
0.01 


50 


100 


(W) 

FIG.  69. 


150 


200 


250 


The  representative  curve  and  its  equation.  —  We  now  draw  a  smooth 
curve  passing  very  near  to  the  points  of  the  plot,  so  that  the  deviations 
of  the  points  from  the  curve  are  very  small,  some  positive  and  some 
negative.  In  Fig.  69,  the  points  seem  to  fall  approximately  ton  a  straight 
line.  This  should  be  tested  by  moving  a  stretched  thread  or  by  sliding 
a  sheet  of  celluloid  with  a  fine  line  scratched  on  its  under  side  among  the 
points  and  noting  that  the  points  do  not  deviate  systematically  from  this 
thread  or  line.  Having  decided  that  a  straight  line  will  approximate 
the  plot,  we  assume  that  an  equation  of  the  first  degree,  E  =  a  +  bW,  will 
approximately  represent  the  relation  between  the  measured  quantities. 
In  this  example  we  may  evidently  assume  that  E  =  bW  since  a  zero  load 
gives  a  zero  elongation. 


124  EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 

The  determination  of  the  cdnstant.  —  We  shall  now  determine  the  con- 
stant b  in  the  equation  E  =  bW.  This  may  be  done  in  several  ways.  The 
three  methods  which  are  generally  employed  are  as  follows : 

I.  Method  of  selected  points.  —  Place  the  sheet  of  celluloid  on   the 
coordinate  paper  so  that  the  scratched  line  passes  through  the  point 
W  =  o,  E  =  o,  and  then  rotate  the  sheet  until  a  good  average  position 
among  the  plotted  points  is  obtained,  i.e.,  until  the  largest  possible  num- 
ber of  points  lie  either  on  the  line  or  alternately  on  opposite  sides  of  the 
line,  in  such  a  manner  that  the  points  below  the  line  deviate  from  it  by 
approximately  the  same  amount  as  the  points  above  it.     Then  note  the 
values  of  W  and  E  corresponding  to  one  other  point  on  this  line,  prefer- 
ably near  the  farther  end  of  the  line.     Thus  we  read  W  =  250,  E  = 
0.0650.     Substituting  these  values  in  the  equation  E  =  bW,  we  have 
0.0650  =  250  b,  and  hence  b  =  0.000260,  and  finally  E  =  0.000260  W. 
Since  the  choice  of  the  "best"  line  is  a  matter  of  judgment,  its  position, 
and  hence  the  value  of  the  constant,  will  vary  with  different  workers  and 
often  with  the  same  worker  at  different  times. 

II.  Method  of  averages.  —  The  vertical  distances  of  the  plotted  points 
from  the  representative  line  are  called  the  residuals;  these  are  the  differ- 
ences between  the  observed  values  of  E  and  the  values  of  E  calculated 
from  the  formula,  or  E  —  Ec,  where  Ec  =  bW',  some  of  these  residuals 
are  positive  and  others  are  negative.     If  we  assume  that  the  "best"  line 
is  that  which  makes  the  algebraic  sum  of  the  residuals  equal  to  zero,  we 
have 

2(E  -  bW]  =o     or    ZE  -  bZW  =  o, 

2E      0.3225 

hence  b  =  ^TFT>  =  —  —    =  0.000261, 

2W       1235 

and  we  may  call  this  an  average  value  of  b.  By  this  method  it  is  no 
longer  necessary  to  shift  the  line  among  the  points  so  as  to  get  an  average 
position. 

III.  Method  of  Least  Squares.  —  In  the  theory  of  Least  Squares  *  it  is 
shown  that  the  best  line  or  the  best  value  of  the  constant  is  that  which 
makes  the  sum  of  the  squares  of  the  differences  of  the  observed  and  cal- 
culated values  a  minimum,  i.e., 

2  (E  -  bW}2  =  minimum. 

Hence  the  derivative  of  this  expression  with  respect  to  b  must  equal  zero, 
or 

^  2  (E  -  bW)z  =  o,     or    S  W(E  -  bW)  =  o, 
or  2WE  -  bZW*  =  o,     and     b  = 

j 
*  See  Bartlett's  "  The  Method  of  Least  Squares,"  or  any  other  book  on  this  theory. 


ART.  70 


THE   STRAIGHT  LINE,  y  =  a  +  bx 


125 


We  form  two  columns,  one  giving  the  values  of  RW  and  the  other  the 
values  of  W2,  and  adding  these  columns,  we  find 

b  =  67.007/255,725  =  0.000262. 

We  may  now  compare  the  results  obtained  by  each  of  the  three 
methods.  For  this  purpose  we  complete  the  table  by  computing  the 
values  of  E  from  the  formulas 

I.  E  =  0.000260  W;    II.  E  =  0.000261  W;    III.  E  =  0.000262  W. 

These  are  marked  Ec*,  Ecu,  Ecm,  in  the  table.  To  discover  how  closely 
the  computed  values  agree  with  the  observed  values  we  form  the  residuals 

A1  =  E  -  Ec\        A11  =  E  -  Ecn,        A111  =  E  -  E™. 

Disregarding  the  signs  of  these  residuals,  we  add  them  and  divide  by 
their  number,  8,  and  find  the  average  residual  to  be  0.00048,  0.00045, 
0.00043,  respectively.  We  also  find  the  sum  of  the  squares  of  the  resid- 
uals to  be  356,  270,  254,  respectively.  We  may  therefore  draw  the  fol- 
lowing conclusions:  all  three  methods  give  good  results;  the  method  of 
Least  Squares  gives  the  best  value  of  the  constant  but  requires  the  most 
calculation;  the  method  of  averages  gives,  in  general,  the  next  best  value 
of  the  constant  and  requires  but  little  calculation;  the  graphical  method 
of  selected  points  requires  the  least  calculation  but  depends  upon  the 
accuracy  of  the  plot  and  the  fitting  of  the  representative  line. 

70.  The  straight  line,  y  =  a  +  bx.  —  For  measuring  the  temperature 
coefficient  of  a  copper  rod  of  diameter  0.3667  in.  and  length  30.55  in.,  the 
following  measurements  were  made.  Here,  C  is  the  temperature  Centi- 
grade and  r  is  the  resistance  of  the  rod  in  microhms. 


c 

' 

C» 

rC 

•V 

r=» 

rcm 

A1 

A" 

A1" 

19.1 

76.30 

364-81 

1.457-33 

76.19 

76.19 

76.26 

-t-o.ii 

+0.1  1 

+0.04 

25.0 

77.80 

625.00 

1,945.00 

77.91 

77.92 

77.96 

—  O.II 

—  0.12 

—  0.16 

30.1 

36.0 

79-75 
80.80 

906  .01 
1296  .00 

2,400.48 
2,908.80 

79-39 
8i.ii 

79-41 
81.14 

79-43 
81.13 

+o.3b 
-0.31 

+0-34 
-0-34 

+0.32 
-0.33 

40.0 

82.35 

1600  .00 

3,294.00 

82.27 

82.31 

82.28 

+0.08 

+0.04 

+0.07 

45-1 

83.90 

2034.01 

3.783-89 

83.75 

83-80 

83.76 

+0.15 

+0.10 

+0.14 

85.10 

2500.00 

4,255-00 

85.18 

85.24 

85.16 

—0.08 

—0.14 

—0.06 

2245.3 

566.00 

9325.83 

20,044  .  50 

1.20 

I.I9           I.  12 

S  -j-  7  =  0.171 

0.170 

0.160 

SAZ=    2852 

2869 

2646 

The  plot  (Fig.  70)  appears  to  approximate  a  straight  line,  so  that  we 
shall  assume  the  relation  r  =  a  +  bC.  We  shall  determine  the  con- 
stants, a  and  b,  by  the  three  methods. 

L  Method  of  selected  points.  —  Use  a  sheet  of  celluloid  to  determine 
the  approximate  position  of  the  best  straight  line,  and  note  two  points 


126 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


on  this  line;  thus,  C  =  20,  r  =  76.45,  and  C  =  48,  r  =  84.60.     Substi- 
tuting these  values  in  the  equation  r  =•  a  +  bC,  we  get 

76.45  =  a  +  20  b    and     84.60  =  a  +  48  b, 
from  which  we  determine 

a  =  70.63    and    b  =  0.291, 
so  that  our  relation  becomes 

r  =  70.63  +  0.291  C. 


85 
84 
83 
82 
81 

80 
(r) 

79 
78 
77 
76 
75 

/ 

/ 

/ 

f 

/ 

/ 

/ 

+ 

/ 

7 

/ 

/ 

£ 

7 

/ 

10       15       20      25      30      85      40      46      50 

FIG.  70. 

II.  Method  of  averages.  —  Since  we  have  to  determine  two  constants, 
we  divide  the  data  into  two  equal  or  nearly  equal  groups,  and  place  the 
sum  of  the  residuals  in  each  group  equal  to  zero,  i.e., 

I,  (r  -  a  -  bC}  =  o    or    Sr  =  na  +  bZC, 

where  n  is  the  number  of  observations  in  the  group.  Thus,  dividing  the 
above  data  into  two  groups,  the  first  containing  four  and  the  second  three 
sets  of  data,  and  adding,  we  get 

314-65  =  4  a  +  1 10.2  b    and    251.35  =  3  a  -f-  135.1  b, 


AST.  70  THE   STRAIGHT  LINE,  y  =  a  +  bx  12? 

from  which  we  determine 

a  =  70-59    and    b  =  0.293, 
so  that  our  relation  becomes 

r  =  70-59  +  0.293  C. 

III.  Method  of  Least  Squares.  —  The  best  values  of  the  constants  are 
those  for  which  the  sum  of  the  squares  of  the  residuals  is  a  minimum,  i.e., 
2  (r  —  a  —  bC)z  =  minimum;  hence  the  partial  derivatives  of  this 
expression  with  respect  to  a  and  b  must  be  zero  ;  thus, 

±Z(r-a-  bC?  =  o,      1  S  (r  -  a  -  6C)2  =  o, 

or      2[2(r-a-  bC]  (-i)j  =  o,        2  [2  (r  -  a  -  bQ  (-Q]  =  o, 
or  Zr  =  an  + 


where  n  is  the  number  of  observations.  We  solve  these  last  two  equations 
for  a  and  b.  (Note  that  these  equations  may  be  formed  as  follows: 
substitute  the  observed  values  of  r  and  C  in  the  assumed  relation  r  =  a 
-f-  bC;  add  the  n  equations  thus  formed  to  get  the  first  of  the  above 
equations;  multiply  each  of  the  n  equations  by  the  corresponding  value 
of  C  and  add  the  resulting  n  equations  to  get  the  second  of  the  above 
equations.) 

We  now  compute  the  values  of  rC,  C2,  ZC,  ZrC,  and  ZC2,  and  substi- 
tute these  in  the  equations  for  determining  a  and  b.  We  thus  get 

566.00  =  7  a  +  245.3  b, 
20,044.50  =  245.3  a  +  9325-83  b, 
from  which  we  determine 

a  =  70.76     and     b  =  0.288, 
so  that  our  relation  becomes 

r  =  70.76  +  0.288  C. 

Comparison  of  results.  —  We  note  that  the  various  results  agree  very 
well  with  the  original  data  and  with  each  other.  We  compute  the  resid- 
uals and  find  that  the  average  residual  is  smallest  by  the  third  method 
and  is  approximately  the  same  by  the  first  two  methods.  The  computa- 
tion necessary  in  applying  the  method  of  Least  Squares  is  very  tedious. 
The  method  of  selected  points  requires  the  fitting  of  the  best  straight  line, 
and  this  becomes  quite  difficult  when  the  number  of  plotted  points  is 
large.  We  shall  therefore  use  the  method  of  averages  in  most  of  the 
illustrative  examples  which  follow. 


128  EMPIRICAL  FORMULAS  — NON-PERIODIC   CURVES         CHAP.  VI 

(II)   FORMULAS  INVOLVING  TWO   CONSTANTS. 

71.  Simple  parabolic  and  hyperbolic  curves,  y  =  ax6.  —  As  stated  in 
Art.  68,  when  the  plotted  points  deviate  systematically  from  a  straight 
line,  a  smooth  curve  is  drawn  so  as  to  pass  very  near  the  points;  the  shape 
of  the  curve  or  a  knowledge  of  the  nature  of  the  experiment  may  give  us 
a  hint  as  to  the  form  of  the  equation  which  will  best  represent  the  data. 

Simple  curves  which  approximate  a  large  number  of  empirical  data 
are  the  parabolic  and  hyperbolic  curves.  The  equation  of  such  a  curve 
is  y  =  ax6,  parabolic  for  b  positive  and  hyperbolic  for  b  negative.  In 
Fig.  710,  we  have  drawn  some  of  these  curves  for  a  =  2  and  b  =  —  2, 
—  I,  —0.5,  0.25,  0.5,  1.5,2.  Note  that  the  parabolic  curves  all  pas 


through  the  points  (o,  o)  and  (i,  a)  and  that  as  one  of  the  variables 
increases  the  other  increases  also.  The  hyperbolic  curves  all  pass  through 
the  point  (i,  a)  and  have  the  coordinate  axes  as  asymptotes,  and  as  one 
of  the  variables  increases  the  other  decreases. 

There  is  a  very  simple  method  of  verifying  whether  a  set  of  data  can 
be  approximated  by  an  equation  of  the  form  y  =  ax*.  Taking  loga- 
rithms of  both  members  of  this  equation,  we  get  log  y  =  log  a  +  b  log  x, 
and  if  x'  =  log  x,  y'  =  log  y,  this  becomes  y'  =  log  a  +  bx't  an  equation 
of  the  first  degree  in  x'  and  y' ;  therefore  the  plot  of  (xr,  y')  or  of  (log  x, 
log  y)  must  approximate  a  straight  line.  Hence, 


ART.  71 


PARABOLIC  AND   HYPERBOLIC   CURVES,  y 


I29 


//  a  set  of  data  can  be  approximately  represented  by  an  equation  of  the 
form  y  =  arc6,  then  the  plot  of  (log  x,  log  y)  approximates  a  straight  line. 

Instead  of  plotting  (log*,  logy)  on  ordinary  coordinate  paper,  we 
may  plot  (x,  y)  directly  on  logarithmic  coordinate  paper  (see  Art.  13). 
We  determine  the  constants  a  and  b  from  the  equation  of  the  straight 
line  by  one  of  the  methods  described  in  Art.  70. 

Example.  The  following  table  gives  the  number  of  grams  S  of  anhy- 
drous ammonium  chloride  which  dissolved  in  100  grams  of  water  makes 
a  saturated  solution  of  6°  absolute  temperature. 

2.43  2.44  2.45  2.46  2.47  2.48  2.49  2 5$ 2.51  2.52  2.53  2.54  2.55  2.56  2.57 


75 
70 
65 
60 

55 

(S) 
50 

45 
40 
35 
30 
25 

^ 

^ 

/.«(? 

/.75 
(log 

1.60 
1.55 
1  50 

/ 

X 

4 

./ 

X 

x 

/x 

X 

<s 

w 

$ 

^ 

2 

' 

^ 

S 

X 

y 

J 

\ 

x 

<^ 

4 

S 

^ 

^ 

x 

x 

1.45 

+ 

*' 

0  280  290  300  310  31 
(* 

0  550  340  350  360  370 

) 

FIG.  716. 


9 

5 

log  0 

log  5 

Scl 

•SV1 

A' 

A" 

273 

29.4 

.4362 

.4684 

29-7 

29-7 

-0-3 

-0-3 

283 

33-3 

.4518 

•5224 

33-2 

33-2 

+0.1 

+0.1 

288 

35-2 

•4594 

•5465 

35  -o 

35-i 

+0.2 

+0.1 

293 

37-2 

.4669 

•5705 

37-0 

37-0 

+0.2 

+0.2 

313 

45-8 

•4955 

.6609 

45-3 

45-3 

+0.5 

+0-5 

333 
353 

55-2 
65.6 

.5224 
•5478 

•7419 
1.8169 

It:? 

ill 

+0.3 

—  O.I 

+0-3 
—  O.2 

373 

77-3 

•5717 

1.8882 

77-9 

78.0 

-0.6 

—  0.7 

2-^-8 

=  0.29 

0.30 

130 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


The  points  (6,  S)  are  plotted  in  Fig.  716.  The  curve  appears  to  be 
parabolic,  i.e.,  of  the  general  form  illustrated  in  Fig.  710.  We  therefore 
plot  (log  6,  log  61)  and  note  that  this  approximates  a  straight  line,  so  that 
we  may  assume 

S  =  aP     or     log  5"  =  log  a  +  b  log  6. 

We  shall  first  determine  the  constants  by  the  method  of  selected 
points.  We  note  two  points  on  the  line  whose  coordinates  are 

Iog0  =  2.445,  log  5  =  1.50    and    log  9  =  2.555,  log  5  =  1.84, 
hence  we  have 

1.50  =  log  a  +  2.445  b, 
1.84  =  logo  +  2.555  b. 

:.     b  =  3.09,    log  a  =  —6.0550  =  3.9450  —  10,    a  =  0.000,000,881. 
.'.     log  51  =  —6.0550  +  3.09  log  0,     or     S  =  0.000,000,881  03-09. 

We  shall  now  determine  the  constants  by  the  method  of  averages. 
We  divided  the  data  into  two  groups  of  four  sets,  and  adding,  we  have 

6.1078  =  4  log  a  +    9.8143  b, 

7.1079  =  4  log  a  +  10.1374  &• 

.'.   b  =  3.09,  log  a  =  —6.0546  =  3.9454  —  10,  a  =  0.000000882. 
/.   log  5  =  —6.0546  +  3.09  log  0    or    S  =  0.000000882  03-09. 

We  complete  the  table  by  computing  S,  the  residuals,  and  the  average 
residual.  The  agreement  between  the  observed  and  computed  values  of 
5  is  quite  close. 

Example.  The  following  table  gives  the  pressure  p  in  pounds  per 
sq.  in.  of  saturated  steam  corresponding  to  the  volume  v  in  cu.  ft.  per 
pound.  (From  Perry's  Elementary  Practical  Mathematics.) 


V 

p 

log* 

log* 

PC 

A 

53-92 

6.86 

1.7318 

0.8363 

6.85 

+0.01 

26.36 

14.70 

1  .4210 

1.1673 

14.69 

+O.OI 

14.00 

28.83 

I  .1461 

i  -4599 

28.85 

—  0.02 

6.992 

60.40 

0.8446 

1.7810 

60.49 

—  O.09 

4.280 

101  .9 

0.6314 

2.0082 

IO2.I 

—  0.2 

2.748 

l63-3 

0.4390 

2.2130 

163.7 

-0.4 

1.853 

250-3 

0.2679 

2-3984 

249.2 

+I.I 

The  points  (v,  p)  are  plotted  in  Fig.  "j\c.  The  curve  appears  to  be 
hyperbolic  on  comparison  with  Fig.  710.  Hence  we  plot  (logy,  log  p) 
and  note  that  this  approximates  a  straight  line,  so  that  we  may  assume 

p  =  atf,     or     log  p  =  log  a  +  b  log  v. 
We  shall  use  the  method  of  averages  to  determine  the  constants  a  and  b. 


ART.  72 


SIMPLE  EXPONENTIAL  CURVES,  y  = 


Dividing  the  data  into  two  groups,  the  first  four  and  the  last  three  sets, 
and  adding,  we  have 

5.2445  =  4  log  a  +  5-I4356, 
6.6196  =  3  log  a  +  1-3383  b. 
.'.     b  =  —1.0662,     log  a  =  2.6822,     a  =  481.1. 
.'.     \ogp  =  2.6822  -  i. 0662  log  v,     or    pu1-0"*  =  481.1. 

We  now  compute  p  and  A  and  note  the  close  agreement  between  the 
observed  and  calculated  values. 


(logv) 
0.2    0.4    0.6    0.8     1.0     1.2    1.4     1. 

6     1.8 

2.4 
2.2 
2.0 

1.8 

^ 

'•'t 

1.4 
1.2 
1.0 
0.8 

9&n  - 

nnn  . 

{ 

nnn  . 

\ 

^L 

\ 

tan 

\ 

r. 

\ 

4 

(P) 

\ 

inn 

\ 

SO 

t 

\ 

nn  - 

u 

\ 

Aft 

\ 

s. 

. 

'v,  n\ 

0  . 

~~~  —  ^ 

s  — 

—  h 

5      10      15     20     25     30     35     40     45     50    55 
(v) 

FIG.  7 ic. 

72.  Simple  exponential  curves,  y  =  aeba>.  —  Other  simple  curves  that 
approximate  a  large  number  of  experimental  results  are  the  exponential 
or  logarithmic  curves.  The  equation  of  such  a  curve  may  be  written  in 
the  form  y  =  a^x,  where  e  is  the  base  of  natural  logarithms;  the  form 
y  =  ab*  is  sometimes  used.  In  Fig.  720,  we  have  drawn  some  of  these 
curves  for  a  =  i  and  b  =—2,  —  i,  —0.5,  0.5,  i,  2.  Note  that  these 
curves  all  pass  through  the  point  (o,  a)  and  have  the  x-axis  for  asymptote. 


EMPIRICAL   FORMULAS  —  NON-PERIODIC   CURVES          CHAP.  VI 


3.0 


2.5 


2.0 


There  is  a  very  simple  method  of  verifying  whether  a  set  of  data  can 
be  approximated  by  an  equation  of  the  form  y  =  a&*.  Taking  logarithms 

of  both  members  of  this  equa- 
tion we  get  log  y  =  log  a  + 
(b  log  e)  x,  and  if  y'  =  log  y, 
this  equation  becomes  y'  = 
log  a  +  (b  log  e)  x,  an  equation 
of  the  first  degree  in  x  and  y' ; 
therefore  the  plot  of  (x,  y'}  or 
of  (x,  log  y)  must  approximate 
a  straight  line.  Hence, 

//  a  set  of  data  can  be  ap- 
proximately represented  by  an 
equation  of  the  form  y  =  ae*1, 
then  the  plot  of  (x,  log  y)  ap- 
proximates a  straight  line. 

Instead  of  plotting  (x,  logj) 
on  ordinary  coordinate  paper, 
we  m?.y  plot  (x,  y}  directly 


1.5 


1.0 


0.5 


(x) 

y  =  ael 
FIG.  72 


0    0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0   on  semilogarithmic  coordinate 

paper  (see  Art.  14).     The  con- 
stants a  and  b  are  determined 
from    the    equation    of    the 
straight  line  by  one  of  the  methods  described  in  Art.  70. 

Example.  Chemical  experiments  by  Harcourt  and  Esson  gave  the 
results  of  the  following  table,  where  A  is  the  amount  of  a  substance  re- 
maining in  a  reacting  system  after  an  interval  of  time  /. 


t 

A 

log/ 

log  A 

AC 

A 

2 

94.8 

0.3010 

.9768 

94-9 

—  O.I 

5 

87.9 

0.6990 

.9440 

87.7 

+0.2 

8 

81.3 

0.9031 

.9101 

81.0 

+0.3 

ii 

74-9 

I  .0414 

.8745 

74.8 

+  0.1 

14 

68.7 

I  .  1461 

.8370 

69.1 

-0.4 

i? 

64  .0 

I  .  2304 

.8062 

63.8 

+0.2 

27 

49-3 

i  -43*4 

.6928 

49-0 

+0.3 

3i 

44.0 

i  .4914 

•6435 

44-i 

—  O.I 

35 

39-i 

i  -5441 

•5922 

39-6 

-0-5 

44 

31-6 

I-643S 

•4997 

31.2 

+0.4 

ZA  -i-  10  =  0.26 

The  points  (/,  A)  are  plotted  in  Fig.  726.  This  curve  appears  to  be 
exponential,  so  that  we  plot  (/,  log  A)  and  (log/,  A);  it  is  seen  that  the 
plot  of  (/,  log  A)  approximates  a  straight  line.  We  may  therefore  assume 
an  equation  of  the  form 

A  =  ae*'     or     log  A  =  log  a  +  (b  log  e)  t. 


ART.  72  SIMPLE  EXPONENTIAL  CURVES,  y  -  o«** 

(I*  0 
0.2      0.4      0.6     0.8      1.0      1.2      1.4      1.6      1.8 


133 


95 
90 
?5 

Of) 

2.00 
1.95 
1.90 
1.85 

1.80 

1.75 

(log  A) 
1.70 

1.65 
1.60 
1.55 
1.50 
1.45 

\ 

•*>! 

\ 

V,, 

'x, 

\ 

\ 

\ 

\ 

s^ 

\ 

\ 

\ 

\\ 

\ 

^ 

\ 

\ 

75 
70 

65 

<^) 
60 

55 
50 
45 
4JS 
35 

30c 

>; 

Si 

\ 

V 

\ 

. 

\ 

X 

\ 

£ 

\ 

\ 

z 

\ 

\ 

r 

A" 

\ 

\1 

•^ 

^ 

^ 

^ 

5 

\ 

S 

Nj 

\ 

\ 

\ 

\ 

^L 

',    \ 
f 

s 

\ 

\ 

•\ 

\ 

\ 

\ 

4 

\ 

V 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

s 

\. 

s 

\ 

\ 

S 

\ 

S 

N\ 

Vx 

\ 

X 

)         5/0/5       20      25       30       35       40       45 

FIG.  726. 

We  shall  use  the  method  of  averages  to  determine  the  constants.     Divid- 
ing the  data  into  2  groups  and  adding,  we  get 

9.5424  =  5  log  a  +  40  (b  log  e}, 
8.2344  =  5  log  a  +  154  (b  log  e): 
:.     b\oge=  —0.0115,    log  a  =  2.0005. 
b  =  —0.0265,  a  =  loo.i,  since  log  e  =  0.4343. 

.-.     log  ,4  =  2.0005  —  0.0115 /,     or    ^4  =  100. i  g-«-<«65«. 

We  now  compute  the  values  of  A  and  the  residuals,  and  note  the  close 
agreement  between  the  observed  and  the  calculated  values  of  A . 


134 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


Example.    The  following  table  gives  the  results  of  measuring  the 
electrical  conductivity  C  of  glass  at  temperature  6°  Fahrenheit. 


0.09 
0.08 
0.07 
0.06 
0.05 

(CO 

0.04 
0.03 
0.02 
0.01 
0 

f 

y.u 

0  n 

/ 

y 

8.8 
o  r 

^ 

/ 

-t 

/ 

0  ft 

/ 

/ 

j 

8.5 

0  * 

i 

/ 

V 

/ 

1 

f 

§ 

/ 

* 

I 

/ 

4 

7 

/ 

^ 

i 

y 

8.1 

o  n 

/ 

Q 

Z 

/ 

/ 

/ 

r 

7.9 
7.8 
7.7 

V 

1 

/ 

I 

A 

1 

x 

/ 

-^ 

"*"*" 

/ 

\ 

/ 

> 

0       60       80'     100     120     140     160      180     20Q~'  22Q    24 

FIG.  72  c. 

0 

C 

log  0 

logC 

C, 

A 

58 

O 

I  •  7634 

—  00 

0.0019 

86 

O.OO4 

i  -9345 

7.6021  —  10 

0.0039 

-fo.oooi 

148 

O.OlS 

2.1703 

8.2553-10 

0.0185 

—  0.0005 

1  66 

O.O29 

2.22OI 

8.4624—10 

0.0292 

—  O.OOO2 

188 

0.051 

2.2742 

8.7076  —  10 

0.0510 

O 

202 

0.073 

2.3054 

8.8633  —  10 

0.0728 

+0.0002 

210 

0.090 

2.3222 

8.9542-10 

0.0891 

—  O.OOIO 

ART.  73 


PARABOLIC  OR  HYPERBOLIC   CURVE,  y  =  a  +  bxn 


135 


In  Fig.  72c,  the  pofnts  (9,  C)  and  (0,  log  C)  are  plotted;  the  latter 
plot  approximates  a  straight  line.  We  may  therefore  assume  the  equa- 
tion 

C  =  ae™,     or     log  C  =  log  a  -f-  (b  log  e)  0. 

We  use  the  method  of  averages  to  determine  the  constants.  Omitting 
the  first  set  and  dividing  the  remaining  data  into  two  groups  of  three 
sets,  we  get 

24.3198  -  30  =  3  log  a  +  400  (b  log  e), 
26.5251  -  30  =  3  log  a  +  600  (b  log  e). 
.'.     bloge  =  o.ono,     logo  =  6.6399  —  10. 
.'.     b  =  0.0253,  a  =  0.000436. 

.'.     log  C  =  6.6399  —  10  +  o.ono  6,     or     C  =  0.00436  e0-0258', 

We  now  compute  the  values  of  C  and  the  residuals  and  note  the  re- 
markably close  agreement  between  the  observed  and  computed  values 
of  C. 

73.  Parabolic  or  hyperbolic  curve,  y  =  a  +  bxn  (where  n  is  known).  — 
In  using  this  equation,  it  is  assumed  that  from  theoretical  considerations 
we  suspect  the  value  of  n.  It  is  evident  that 

If  a  set  of  data  can  be  approximately  represented  by  an  equation  of  the 
form  y  =  a  +  bxn,  where  n  is  known,  then  the  plot  of  (xn,  y}  approximates 
a  straight  line. 

Example.  A  small  condensing  triple  expansion  steam  engine  tested 
under  seven  steady  loads,  each  lasting  three  hours,  gave  the  following 
results;  I  is  the  indicated  horse-power,  w  is  the  number  of  pounds  of 
steam  used  per  hour  per  indicated  horse-power.  (From  Perry's  Ele- 
mentary Practical  Mathematics.) 


/ 

w 

wl 

wa 

A 

36.8 

12.  S 

460.0 

12.6 

—o. 

31.5 

I2.9 

406.4 

12.8 

+o. 

26.3 

I3-I 

344.5 

13.0 

+0. 

21  .O 

13-3 

279-3 

13-4 

—  o. 

IS.8 

I4.I 

222.8 

14.0 

+  0. 

12.6 

14-5 

182.7 

14.6 

—  o. 

8-4 

I6.3 

136.9 

16.1 

+0. 

"SA  -i-  7  =  o.n 

Fig.  73«  gives  the  plot  of  (I,  w).  This  is  not  a  straight  line.  But  if 
we  plot  (/,  wl),  i.e.,  the  total  weight  of  steam  used  per  hour  instead  of 
the  weight  per  indicated  horse-power,  we  find  that  this  plot  approximates 
a  straight  line.  Hence,  we  may  assume  the  linear  relation  wl  =  a  +  bl. 
This  relation  may  also  be  written  w  =  b  +  a/ 1,  so  that  the  plot  of  (l/J,  w) 
also  approximates  a  straight  line.  We  use  the  method  of  averages  to 


136 


EMPIRICAL   FORMULAS  —  NON-PERIODIC   CURVES 


CHAP.  VI 


determine  the  constants.     Dividing  the  data  into  two  groups,  the  first 

three  and  last  four  sets,  and  adding,  we  have 

1210.9  =  3  a  +  94.66, 

821.7  =  4  a  +  57-86. 

.*.     b  =  1 1. 6,     a  =  37.8. 

.'.     wl  =  37.8  -f-  II.67,    or    w=ii.6  +  ^y— 
We  now  compute  the  values  of  w  and  the  residuals. 


16 


15 


14 


13 


/ 

450 
400 
350 
300 
250  (™  I) 

200,. 
150 
100 
50 
n 

[ 

/ 

I 

/ 

\\ 

V 

/ 

\\ 

t 

/ 

\\ 

/ 

\\ 

^ 

\ 

/\ 

9 

\ 

/ 

\. 

/ 

\ 

/ 

> 

ft- 

+• 

/ 

V 

> 

/ 

/ 

"v 

/ 

4 

V 

/ 

x 

<f- 

^ 

<t 

^^^- 

-^, 

^^ 

10        15       20       25       30       35       40 

FIG.  730- 

Example.  For  a  parachute  or  flat  plate  falling  in  air  we  have  the 
following  observations;  v  is  the  velocity  in  ft.  per  sec.  and  p  is  the  pressure 
in  pounds  per  sq.  in. 


V 

p 

V- 

PC 

A 

7.87 

0.2 

61.94 

0.187 

—  0.013 

11.50 

0.4 

132.25 

0.401 

+0.001 

16.40 

0.8 

268.96 

0.8IS 

—0.015 

22.60 

1.6 

510.76 

1.548 

+0.052 

32.80 

3-2 

1075.84 

3.260 

—0.060 

SA  -r-  5  =  0.028 


ART.  74 


HYPERBOLIC   CURVE,  y 


a  +  bx 


137 


In  Fig.  73&,  we  have  plotted  (v,  p}.  It  is  surmised  that  for  low  veloci- 
ties, the  pressure  and  the  square  of  the  velocity  are  linearly  related,  i.e., 
p  =  a  +  bv2.  We  verify  this  by  plotting  (v2,  p)  and  noting  that  this 
approximates  a  straight  line.  We  use  the  method  of  averages  to  deter- 

(v*) 
0       tOO    200     300     400    500     606     700    800     900    1000  1100 


8.5 
3.0 
2.5 

G?)  2-0 

1.5 
J.O 
0.5 

QL 

/ 

s 

/ 

£ 

/ 

F 

/ 

/ 

^ 

/ 

s 

/ 

/ 

/ 

\< 

y 

^ 

/ 

/ 

sS 

rf 

^ 

^ 

^ 

/ 

s 

s? 

sf 

•tf 

f 

s 

/• 

/ 

/t 

,* 

^A 

# 

^ 

•P 

I         5         10        15       20       25       30       35 
fe) 

FIG.  736. 


mine  the  constants.     Dividing  the  data  into  two  groups,  the  first  three 
and  the  last  two  sets,  and  adding,  we  have 

1.4  =  3  a  +    463.156, 
4.8  =  2  a  +  1586.60  b. 

.*.     b  =  0.00303     and    a  =  —  o.ooin. 
.'.     p  =  —  o.ooiu  +0.00303^. 

We  may  with  good  approximation  take  a  =  o,  so  that  p  =  0.00303  0s,  i.e., 
the  pressure  varies  directly  as  the  square  of  the  velocity. 


74.   Hyperbolic  curve,  y 


or  -  =  a  +  bx.  — This  equation 


+  bx*       y 

represents  the  ordinary  hyperbola  with  asymptotes  x  =  —a/b  and  y  = 
i/b,  as  illustrated  in  Fig.  740  for  values  of  a  =  0.2,  b  —  0.2;  a  =  o.i, 
b  =  0.2;  a  =  —o.i,  b  =0.2;  a  =  —0.2,  b  =  0.2.  Quite  a  large  number 
of  experimental  results  may  be  represented  by  an  equation  of  this  type. 


138  '  EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 

The  equation  may  also  be  written  in  the  form    -  =  b  +  -,  so  that  the 

plots  (x,  -)  and  (-,  -1  approximate  straight  lines.     Hence, 

If  a  set  of  data  can  be  approximately  represented  by  an  equation  of  the 

X  OC  I         ^C\  /I        I  \ 

form  y  —  — ; — r-,  or  -  =  a  +  bx  then  the  plot  of  [x,  -I  or  of  I  -  ,  - 1  approxi- 
a  +  bx '      y  \     yJ          \x '  yj   l 

mates  a  straight  line. 
10 


FIG.  740.     y 


a  +  bx 


Example.  From  a  magnetization  or  normal  induction  curve  for 
iron  we  find  the  following  data;  H  is  the  number  of  Gilberts  per  cm.,  a 
measure  of  the  field  intensity,  and  B  is  the  number  of  kilolines  per  sq.  cm., 
a  measure  of  the  flux  density. 


H 

B 

H/B 

Bc 

A 

2-S 

3-5 

0.714 

7-97 

3-o 

5-o 

o.6ob 

8.78 

3-i 

7-5 

0-4I3 

8.91 

3-8 

10.  0 

0.380 

9-8 

+0.2 

7.0 

*2-S 

0.560 

12.4 

+0.! 

9-5 

13-5 

0.703 

13-6 

—  O.I 

"•3 

14.0 

0.808 

14.0 

0 

*7-5 

*s-° 

1.17 

*S-i 

—  O.I 

3*-S 

16.0 

1.97 

16.2 

—  0.2 

45-0 

16.5 

2.72 

16.7 

—  0.2 

64.0 

17.0 

3-76 

17.0 

O 

95-o 

*7-S 

5-43 

17-3 

+0.2 

2A  -s-  9  =  0.12 

In  Fig.  74&,  (H,  B}  is  plotted.     The  curve  appears  to  be  of  the  type 
illustrated  in  Fig.  740.     Furthermore,  an  important  quantity  in  the 


ART.  74 


HYPERBOLIC   CURVE,  y  = 


+  bx 


139 


theory  of  magnetization  is  the  reluctivity  H/B,  and  if  we  plot  (H,  H/B) , 
we  note  that  this  plot  approximates  a  straight  line  for  values  of  H  >  3.1. 
(We  may  similarly  introduce  the  permeability,  B/H,  and  note  that  the 
plot  of  (B/H,  B)  approximates  a  straight  line.)  Hence,  we  assume  a 

TT 

relation  of  the  form   —  =  a  +  bH.     Using  the  method    of  averages, 


100 


FIG.  74&. 

omitting  the  first  three  values  of  H,  and  dividing  the  remaining  data 
into  two  groups  containing  five  and  four  sets  respectively,  we  get  the 
equations  . 

3.621  =  5  a  +  49.1  b, 
13.88  =  4  a  +  235.5  b. 
/.     b  =  0.0560,    a  =  0.174. 

/.     -=  =  0.174  +  0.0560  H    or    B  =  r-^ — z— :=• 

B  0.174  +  0.0560  H 

We  now  compute  B  and  the  residuals  and  note  the  close  agreement 
between  the  observed  and  computed  values. 


140 


EMPIRICAL  FORMULAS  —  NON-PERIODIC  CURVES         CHAP.  VI 


(HI)   FORMULAS  INVOLVING  THREE  CONSTANTS. 

75.  The  parabolic  or  hyperbolic  curve,  y  =  axb  +  c.  —  It  is  often  im- 
possible to  fit  a  simple  equation  involving  only  two  constants  to  a  set  of 
data.  In  such  cases  we  may  modify  our  simple  equations  by  the  addition 
of  a  term  involving  a  third  constant.  Thus  the  equation  y  =  ajc*  may  be 
modified  into  y  =  ax6  -\-  c.  If  b  is  positive,  the  latter  equation  repre- 
sents a  parabolic  curve  with 
intercept  c  on  OF;  if  b  is 
negative,  the  equation  rep- 
resents a  hyperbolic  curve 
with  asymptote  y  =  c.  In 
Fig.  750,  we  have  sketched  the 
curves  y  =  2  ^°-5,  y  =  2  x0-5  +  2, 
y  =  2  x~°-5,  y  =  2  x~°-5  +  2  to 
illustrate  the  relation  of  the 
simple  types  to  the  modified 
types. 

In  Art.  71  it  was  shown 
that  if  we  suspect  a  relation 
of  the  form  y  =  ax11,  we  can 
To  verify  this  by  observing 
whether  the  plot  of  (log  x, 
log  y)  approximates  a  straight 
line.  Now  the  form  y  =  ax11 
+  c  may  be  written  log  (y  —  c}  =  log  a  +  b  log  x,  so  that  the  plot  of 
(log  x,  log  (y  —  c))  would  approximate  a  straight  line.  To  make  this 
test  we  shall  evidently  first  have  to  determine  a  value  of  c.  We  might 
attempt  to  read  the  value  of  c  from  the  original  plot  of  (x,  y}.  In  the 
parabolic  case  we  should  have  to  read  the  intercept  of  the  curve  on  OY, 
but  this  may  necessitate  the  extension  of  the  curve  beyond  the  points 
plotted  from  the  given  data,  a  procedure  which  is  not  safe  in  most  cases. 
In  the  hyperbolic  case,  we  should  have  to  estimate  the  position  of  the 
asymptote,  but  this  is  generally  a  difficult  matter. 

The  following  procedure  will  lead  to  the  determination  of  an  approxi- 
mate value  of  c  for  the  equation  y  =  ax*  +  c.  Choose  two  points  (xi,  yi) 
and  fa,  yd  on  the  curve  sketched  to  represent  the  data.  Choose  a  third 
point  (x3,  yd  on  this  curve  such  that  x3  =  ^XiX2,  and  measure  the  value 
of  y3.  Then,  since  the  three  points  are  on  the  curve,  their  coordinates 
must  satisfy  the  equation  of  the  curve,  so  that 

yi  —  axib  +  c,     yz  =  axzb  +  c,     y3  =  ax3b  +  c. 


75 


PARABOLIC   OR  HYPERBOLIC   CURVE,  y  =  a*6  +  c 


141 


Now,  since  #3  =  Vjc^. 
therefore  x3b  =  Vxibxzb, 

or  y,  -  c 

and  therefore 


and 


=  Vaxf  • 


-  c), 


=     yiy*  - 


-  2  y3 


It  is  evident  that  the  determination  of  c  is  partly  graphical,  for  it 
depends  upon  the  reading  of  the  coordinates  of  three  points  on  the  curve 
sketched  to  represent  the  data.  The  curve  should  be  drawn  as  a  smooth 
line  lying  evenly  among  the  points,  i.e.,  so  that  the  largest  number  of  the 
plotted  points  lie  on  the  curve  or  are  distributed  alternately  on  opposite 
sides  and  very  near  it, 

Having  determined  a  value  for  c,  we  plot  (log  x,  log  (y  —  c)).  If  this 
plot  approximates  a  straight  line,  the  constants  a  and  b  in  the  equation 
log  (y  ~  c)  =  log  a  -\-  b  log  x  may  then  be  determined  in  the  ordinary 
way. 

Example.  In  a  magnetite  arc,  at  constant  arc  length,  the  voltage  V 
consumed  by  the  arc  is  observed  for  values  of  the  current  i.  (From 
Steinmetz,  Engineering  Mathematics.)  , 


i 

V 

V  -  30.4 

log  (V-  30.4) 

logi 

ve 

A 

o.S 

1  60 

129.6 

.1126 

9.6990  —  10 

158.8 

+  1-2 

i 

1  2O 

89.6 

•9523 

o.oooo  —  10 

120.8 

-0.8 

2 

94 

63.6 

•8035 

0.3010  —  10 

94.0 

0 

4 

75 

44-6 

.6493 

0.602  1  —  10 

75-i 

—  O.I 

8 

62 

31-6 

•4997 

0.9031  —  10 

61.9 

+0.1 

12 

56 

25.6 

.4082 

i  .0792  —  10 

56.0 

0 

We  plot  (i,  V)  and  note  that  the  curve  appears  hyperbolic  with  an 
asymptote  V  —  c,  and  hence  we  assume  an  equation  of  the  form  V=  a*6  +  c. 
To  verify  this  we  must  first  determine  a  value  for  c.  Choose  two  points 
on  the  experimental  curve;  in  Fig.  756,  we  read  ii  =  0.5,  FI  =  160  and 
t2  =  12,  F2  =  56.  Choose  a  third  point  such  that  is  =  Vi^  =  V6  = 
2.45,  and  measure  Vs  =  88.  Then 

F^z  -  F»2       =  (160)  (56)  -  (88)2    =  1216 
F!  +  Vt  -  2  F3       160  +  56-2  (88)         40 


30.4. 


Now  compute   the  values  of    V  —  30.4  and   log   ( V  —  30.4)   and  plot 
(log  i,  log  (V  —  30.4)).     This  last  plot  approximates  a  straight  line  so 
that  the  choice  of  the  equation  V  =  aib  +  c  is  verified. 
To  determine  the  constants  in  the  equation 

log  (V  -  30.4)  =  log  a  +  b  log  i, 


142 


EMPIRICAL   FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


we  use  the  method  of  averages,  dividing  the  data  into  two  groups  of  three 
sets  each,  and  find 

5.8684  =  3  log  a, 
4-5572  =  3  log  a  +  2.58446. 
b  =  —0.507,     log  a  =  1.9561,     a  =  90.4. 
.*.     log  (V  -  30.4)  =  1.9561  -  0.507  log*,  or  V  =  30.4  +  90.4 i-0807. 

Finally,  we  compute  the  values  of  V  and  the  residuals. 

(log 


irl 

6   9. 

7    9. 

8    9. 

9    0. 

0    0. 

1    0. 

2    0. 

3    0 

4    0 

5    0. 

6    0. 

7    0 

8   0. 

d   i. 

0    1. 

1 

160 

p  n 

\ 

r>   J 

\ 

^ 

2   0 

\ 

^ 

^ 

(V) 
100 

\ 

^ 

v* 

V,- 

i 
I 

/  S"' 

on 

\ 

c 

\ 

tfl 

'** 

on 

\ 

S 

^ 

r  ff 

X 

V 

fr  7 

"^ 

^ 

f  T 

€0 

50 

.1 

^ 

•^-^ 

•—  —  * 

~—~, 

-•  — 

~   ^ 

—     .. 

- 

^ 

/.4 

) 

. 

>     ; 

t     < 

\      i 

( 

'«' 

I 

? 

y     / 

y    / 

/      / 

2 

FIG.  75&. 

76.  The  exponential  curve,  y  =  ae6""  +  c.  —  The  simple  exponential 
equation  y  =  ae?z  may  have  to  be  modified  into  y  =  ae?>x  +  c  in  order  to 
fit  a  given  set  of  data.  In  the  latter  curve,  the  asymptote  is  y  —  c. 
In  Fig.  760,  we  have  sketched  the  curves  y  =  2  e0-11,  y  =  2  eP-lx+  i» 
y  =  2  e-°lx,  ;y  =  2  e-^-1*  +  I. 

In  Art.  72  it  was  shown  that  if  we  suspect  a  relation  of  the  form 
y  =  ae*x,  we  can  verify  this  by  observing  whether  the  plot  of  (x,  logy) 
approximates  a  straight  line.  Now  y  =  aePx  +  c  may  be  written 
log  (y  —  c)  =  log  a  +  (b  log  0)  x,  so  that  the  plot  of  (x,  log  (y  —  c)) 
would  approximate  a  straight  line.  Evidently  we  shall  first  have  to 
determine  a  value  for  c.  We  proceed  to  do  this  in  a  manner  similar  to 
that  employed  in  Art.  75.  Choose  two  points  (xi,  y\)  and  (x%,  y%)  on 


ART.  76 


THE   EXPONENTIAL   CURVE  y  =  ae6*  +  c 


143 


the  curve  sketched  to  represent  the  data,  and  then  a  third  point  (x3,  ys)  on 
this  curve  such  that  x3  =  \  (x\  +  #2)  and  measure  the  value  of  y3.  Since 
the  three  points  are  on  the  curve. 

yi  =  ae6*1  +  c,         y2  =  ae6*  +  c,         ;y3  =  ae6*3  +  c, 


or   log 


=  (6  log  c)  *!,  log 


(If)  3 
2 


— ~  =  (6  log  e)  *2(  log  *_±  =  (6  log  e}  x3. 


» 

FIG.  760.    y  =  ae6*  +  c 


9     10 


Now,  since 
therefore 

and      log  - 


(b  log  e)  x3 


=  \  (xi  +  xz), 

[(b  log  e)  Xi  +  (b  log  e)  x*], 


-  c    y*  -c 
a      '      a 


Hence      y3  —  c  =  V(yx  —  c)  (yz  —  c),     and     c  = 


-  2  y3 


If  the  data  are  given  so  that  the  values  of  x  are  equidistant,  i.e.,  so 
that  they  form  an  arithmetic  progression,  we  may  verify  the  choice  of 
the  equation  y  =  aebx  +  c  and  determine  the  ^constants  a,  b,  and  c  in  the 
following  manner.  Let  the  constant  difference  in  the  values  of  x  equal  In. 
If  we  replace  x  by  x  +  h,  we  get  y'  =  ae?(x+K)  +  c,  and  therefore,  for  the 
difference  in  the  values  of  y, 

&y  =  y'  -  y  =  ae?>(I+V  -  a(*x  =  ad31  (ef>h  -  l), 
and  log  Ay  =  log  a  (&h  —  i)  +  (b  log  e)  x. 


144 


EMPIRICAL   FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


This  last  equation  is  of  the  first  degree  in  x  and  log  Ay  so  that  the  plot  of 
(x,  log  Ay)  is  a  straight  line.  To  apply  this  to  our  data,  we  form  a 
column  of  successive  differences,  Ay,  of  the  values  of  y,  and  a  column  of 
the  logarithms  of  these  differences,  log  Ay,  and  plot  (x,  log  Ay) ;  if  the 
equation  y  =  aefi1  -f-  c  approximates  the  data,  then  this  last  plot  will 
approximate  a  straight  line.  We  may  then  determine  b  log  e  and 
log  a  (^h  —  i)  and  hence  a  and  b  in  the  ordinary  way,  and  finally  find  an 
average  value  of  c  from  2y  =  aSe6*  +  nc,  where  n  is  the  number  of  data. 
Example.  In  studying  the  skin  effect  in  a  No.  oooo  solid  copper 
conductor  of  diameter  1.168  cm.,  Kennelly,  Laws,  and  Pierce  found  the 
following  experimental  results;  F  is  the  frequency  in  cycles  per  second, 
L  is  the  total  abhenrys  observed. 


F 

L 

L-  si,  860 

log  (L  —  51,860) 

L. 

A 

60 

53.912 

2052 

3.3122 

53.952 

—  40 

306 

53.767 

1907 

3  •  2804 

53.668 

+99 

888 

53,143 

1283 

3.1082 

53,140 

+  3 

1600 

52,669 

809 

2.9079 

52,699 

-30 

2040 

52,499 

639 

2.8055 

52,506 

-   7 

3065 

52,215 

355 

2.5502 

52,212 

+  3 

3950 

52,082 

222 

2-3464 

52,068 

+  14 

5000 

5L965 

I°5 

2.0212 

5i,972  . 

-   7 

In  Fig.  76^,  the  points  (F,  L)  are  plotted;  the  curve  appears  to  be 
exponential  with  an  asymptote  L  =  c.  We  shall  try  to  fit  the  equation 
L  =  ae^F  +  c.  First  determine  an  approximate  value  for  c  by  choosing 
two  points  on  the  experimental  curve,  FI  =  875,  LI  =  53,140,  and  F2  = 
5000,  L2  =  51,980,  and  a  third  point  Fs  =  |  (Fi  +  F2)  =  2938,  L3  = 

52,250.    Then  c  =         *   ' ^-  =  51,860.    Now  compute  (L  —  51,860) 

Li\  -+-  L,z        2  1>3 

and  log  (L  —  51,860),  and  plot  (F,  log  (L  —  51,860));  this  plot  approxi- 
mates a  straight  line,  thus  verifying  the  choice  of  equation.  We  deter- 
mine the  constants  in  the  equation  log  (L  —  51,860)  =  log  a  +  (b  log  e)  F 
by  the  method  of  averages.  Dividing  the  data  into  two  groups  of  four 
sets  each  and  adding,  we  have 

12.6087  =  4  log  a  +  2854  b  log  e, 
9.7233  =  4  log  a  -f  14,055  b  log  e. 


and 


or 


bloge  =  -0.0002576,     log  a  =  3.3360, 

b  =  —0.0005931,     a  =  2168. 
iog  (L  -  51,860)  =  3.3360  -  0.0002576  F, 

L    =    51, 860  +  2168  ^-0005931  F 


ART.  77 


THE   PARABOLA,  y  =  a+bx+cx> 


145 


We  now  compute  L  and  the  residuals,  and  note  the  close  agreement 
between  the  observed  and  computed  values  except  for  the  first  two  values 
of  F.  If  we  omit  these  two  values  in  computing  a  and  b,  these  constants 
have  slightly  different  values,  but  the  agreement  between  the  observed 
and  computed  values  of  L  is  about  the  same. 


54,000 
53,800 
53,600 
53,400 
53,200 
53.000 
52.800 
52,600 
52,400 
52.200 
S  2,000 
51.800 

L 

«  4 

X 

\ 

3.2 
3.1 

o  n 

\ 

\ 

\ 

\ 

V 

\ 

+ 

\ 

\ 

N 

^ 

\ 

\ 

2.9 
2.8 

J 

2.6^ 

2.5 
2  4 

\ 

^ 

\ 

\ 

£f- 

^ 

\ 

2> 

1 

^ 

V 

\ 

y>t 

H<" 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

s 

s^ 

s 

2.9 
2.2 
2.1 

Ik' 

^ 

X 

^ 

\ 

\ 

^* 

—  . 

V 

NS 

• 

\ 

\ 

1,000            2,000            3,000            4.000          5.01 
(F) 

FIG.  766. 

77.  The  parabola,  y  =  a  +  bx  +  ex2.  —  The  equation  of  the  straight 
line  y  =  a  +  bx  may  be  modified  by  the  addition  of  a  term  of  the  second 
degree  to  the  form  y  =  a  +  bx  +  cxz.  This  is  the  equation  of  the  ordi- 
nary parabola.  We  may  verify  whether  this  equation  fits  a  set  of  experi- 
mental data  by  one  of  the  following  methods. 


146 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES          CHAP.  VI 


(l)    Choose   any   point    (**,  yk)    on    the   experimental    curve;    then 
=  o,  +  bxk  +  cXk2,  and 

y-yk  =  b(x-xk)+c(x2-xk2),     or    ?~ 


x  —  xk 


This  last  equation  is  of  the  first  degree  in  x  and  ^ — ^-  so  that  the  plot  of 

(x, 1  will  approximate  a  straight  line. 

\     x  —  xk/ 

(2)  If  the  values  of  x  are  equidistant,  i.e.,  if  they  form  an  arithmetic 
progression,  with  common  difference  h,  then  if  we  replace  x  by  x  +  h  in 
the  equation,  we  get  y'  =  a  +  b  (x  +  7z)  +  c  (x  +  h)2  and  Ay  =  y'  —  y 
=  (bh  +  ch2)  +  2  chx.  This  last  equation  is  of  the  first  degree  in  x  and 
Ay,  so  that  the  plot  of  (x,  Ay)  will  approximate  a  straight  line. 

Hence,  if  a  set  of  data  may  be  approximately  represented  by  the  equation 

y  =  a  +  bx  +  ex2,  then  (i)  the  plot  of  (x,  y  ~  yk),  where  (xk,  yk)  are  the 

\     x  —  xk/ 

coordinates  of  any  point  on  the  experimental  curve,  will  approximate  a 
straight  line,  or  (2)  the  plot  of  (x,  Ay),  where  the  Ay's  are  the  differences  in 
y  formed  for  equidistant  values  of  x,  will  approximate  a  straight  line. 

The  following  examples  will  illustrate  the  method  of  determining  the 
constants. 

Example.  In  the  following  table,  6  is  the  melting  point  in  degrees 
Centigrade  of  an  alloy  of  lead  and  zinc  containing  x  per  cent  of  lead. 
(From  Saxelby's  Practical  Mathematics.) 


i 

e 

*  -  36.9 

0-l8l 

0-i8i 
*-j6.9 

6C 

A 

87.5 

292 

50.6 

III 

2.20 

295 

-3 

84.0 

283 

47-i 

102 

2.17 

285 

77-8 

270 

40.9 

89 

2.18 

268 

+  2 

63-7 

235 

26.8 

54 

2.01 

234 

+1 

46.7 

i97 

9-8 

16 

1.63 

199 

—  2 

36.9 

181 

0 

0 

182 

—  I 

In  Fig.  770,  we  have  plotted  (x,  6).  We  shall  try  to  fit  an  equation 
of  the  form  e  =  a  +  bx  +  ex2  to  the  data.  To  verify  this  choice,  observe 
that  the  curve  passes  through  the  point  xk  =  36.9,  6k  =  181,  and  plot  the 

(/\    _  o  T  \ 
x,  —          — J  ;  this  last  plot  approximates  a  straight  line.     (In 

plotting  the  ordinates  for  the  straight  line  a  scale  unit  ten  times  as  large 
as  that  used  for  the  ordinates  of  the  experimental  curve  has  been  used: 
any  further  increase  in  the  scale  unit  would  simply  magnify  the  devia- 


ART.  77 


THE  PARABOLA,  y  =  a  +  bx  +  c*» 

e  -  181 


14? 


f\      _       _  Q  _ 

tfons.)     We  may  now  assume  the  relation  —  —  —  —  =a'  +  b'x,  and  use 

the  method  of  averages  to  determine  the  constants.     Dividing  the  data 
into  two  groups  of  three  and  two  sets  respectively  and  adding,  we  get 

6-55  =  3  a'  +  249.3  bf, 
3.64  =  2  a'  +  110.46'. 

.'.  b'  =  0.0130,  a'  =  1.  10. 
_    =  1.  10  +  0.0130  x,  or  0  =  141.4  +  0.620*  +  0.0130  x*. 


We  now  compute  6  and  the  residuals. 


300 

290 
280 
270 
260 
250 

(e) 

240 
230 
220 
210 
200 
190 
180 


40 


50 


60 
(x) 

FIG.  770. 


70 


Example.  The  following  table  gives  the  results  of  the  measure- 
ments of  train  resistances;  V  is  the  velocity  in  miles  per  hour,  R  is  the 
resistance  in  pounds  per  ton.  (From  Armstrong's  Electric  Traction.) 


148 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


V 

R 

AK 

V> 

R, 

A 

20 

5-5 

3-6 

400 

5-70 

—  O.2O 

40 

9-i 

5-8 

i,  600 

9.08 

+O.O2 

60 

14-9 

7-9 

3,600 

14.82 

+0.08 

80 

22.8 

i°-5 

6,400 

22.86 

—  O.O6 

IOO 

33-3 

12.7 

10,000 

33-22 

+0.08 

120 

46.0 

14,400 

45-90 

+0.10 

2  420 

131-6 

36,400 

In  Fig.  776,  the  plot  of  (V,  R)  appears  to  be  a  parabola,  R  =  a  +  bV 
+  cF2.  Since  the  values  of  V  are  equidistant,  we  shall  verify  our  choice 
of  equation  by  a  plot  of  (V,  A.R) ;  this  last  plot  approximates  a  straight 
line.  We  may  therefore  assume  AR  =  (bh  +  eft2)  -\-2chV,  where  h  =  20. 


50 
45 
40 
35 
30 

(R)25 
20 
15 
10 


20 


40 


80 


100 


120 


(AP 


FIG. 


We  determine  the  constants  in  this  last  equation  by  the  method  of 
averages,  using  the  five  sets  of  values  of  V  and  A.R.  Dividing  these  data 
into  two  groups  of  three  and  two  sets  respectively  and  adding,  we  get 

17.3  =  3(6&  +  c#)  +  120  (2  eft), 
23.2  =  2  (bh  -f  eft2)  +  1 80  (2  eft). 

/.     2  eft  =  0.117,     bh  +  eft2  =  1.08. 

.'.     c  =  0.0029,     b  —  —0.004. 

/.     R  =  a  -  0.004  V  +  0.0029  F2. 


AXT.  78 


THE  HYPERBOLA,  y 


a  +  bx 


149 


We  determine  a  by  substituting  the  six  sets  of  values  of  V  and  R,  and 
summing,  thus 

2R  =  6  a  -  0.004  S  V  +  0.0029  S  F8, 
or  131.6  =  60  —  0.004  (42°)  +  0.0029  (36,400), 

and  therefore  a  =  4.62. 

Hence,  finally,          R  =  4.62  —  0.004  V  +  0.0029  F2. 

We  now  compute  the  values  of  R  and  the  residuals;  the  agreement 
between  the  observed  and  calculated  values  of  R  is  very  close. 


78.   The  hyperbola,  y  =  — — j—  +  c. 
a  *f-  o;c 


tion  of  the  equation  y 


This  equation  is  a  modifica- 
discussed  in  Art.  74.     In  the  latter 


a  +  bx 
equation,  x  =  o  gives  y  =  o,  while  in  the  former,  x  =  o  gives  y  =  c.     We 

JC 

may  verify  whether  the  equation  y  =  —  j—r — f-  c  fits  a  set  of  experi- 


a-\-bx 
mental  data  as  follows.     Choose  any  point  (xk, 

curve;  then  yk  =       ,*\     +  c,  and 
*(*- 


on  the  experimental 


(a  +  foe)  (a  + 


(a  +  to)  +     (a  + 


This  last  equation  is  of  the  first  degree  in  x  and  —    — -,  so  that  the  plot 

of  (x,  — — — -j  will  approximate  a  straight  line. 

Hence,  if  a  set  of  data  may  be  approximately  represented  by  the  equation 
y  =  *  ,  +  c,  the  plot  of  (x,  ^—  — V  where  (xk,  yk)  are  the  coordinates 

of  a  point  on  the  experimental  curve,  will  approximate  a  straight  line. 

Example.  The  following  table  gives  the  results  of  experiments  on 
the  friction  between  a  straw-fiber  driver  and  an  iron  driven  wheel  under 
a  pressure  of  400  pounds;  y  is  the  coefficient  of  friction  and  jc  is  the  slip, 
per  cent.  (From  Goss,  Trans.  Am.  Soc.  Mech.  Eng.,  for  1907,  p.  1099.) 


X 

y 

x  —  0.65 

y-o.i29 

y  -  0.129 

* 

y» 

0.65 

0.129 

0 

0 

0.129 

0.129 

0.87 

0.217 

O.22 

0.088 

•50 

0-253 

0.228 

0.88 

0.228 

0.23 

0.099 

•32 

0.256 

0.232 

0.90 

0.234 

0.25 

0.105 

.38 

0.264 

0.238 

o-93 

0.275 

0.28 

0.146 

.92 

0.274 

0.248 

1.16 

0.318 

0.51 

0.189 

•70 

0.326 

0.304 

i.  80 

0.400 

0.271 

•25 

0-394 

0.388 

2.12 

0.410 

1-47 

0.281 

5-23 

0.410 

0.411 

3-00 

0-435 

2-35 

0.306 

7.68 

0-435 

0.451 

150 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


In  Fig.  78  we  have  plotted  the  points  (x,  y);  the  experimental  curve 

X 

appears  to  be  an  hyperbola  with  an  equation  of  the  form  y  =  — |-  c. 

To  verify  this  we  note  the  point  x  F=  0.65,  y  =  0.129  on  the  curve,  and 
plot  the  points  Ix,  ^—    f~^-\     This  last  plot  approximates  a  straight 

line.     We  may  therefore  assume  the  relation  — —  =  a  +  bx,  and 

y  —  0.129 

0.5 


0.4 


0.3 


(y) 


0.2 


0.1 


0.5 


2.5 


FIG.  78. 

we  shall  determine  the  constants  by  the  method  of  averages.  As  the 
first  three  points  do  not  lie  very  near  this  straight  line,  we  shall  use  only 
the  last  five  sets  of  data,  and  dividing  these  into  two  groups  of  three  and 
two  sets  respectively  and  adding,  we  get 

8.87  =  3  a  +  3.89  b, 

12.91  =  2  a  +  5.12  b. 

.:     b  =  2.77,    a  =  —0.64. 

x  —  0.65  x  —  0.65 

—  =  —0.64  +  2.77  x     or     y  =  —  — ^ — h  0.129. 

y  -  o  129  2.77  x  -  0.64 

If  we  had  used  all  eight  points  in  determining  the  constants,  we  should 
have  obtained 

9.12  =  4  a  +  3.586, 

19.86  =  4  a  +  8.08  b. 

:.     b  =  2.39,     a  =  0.14. 


x  ~  0.65 


=•  0.14  +  2.39  x     or 


x  -  0.65 
2.39  x  +  0.14 


0.129. 


ART.  79 


THE   LOGARITHMIC   CURVE,  log  y  =a  +  bx  +  cy? 


We  have  computed  both  y  and  y'  and  note  that  the  agreement  with 
the  observed  values  is  probably  as  close  as  could  be  expected. 

79.  The  logarithmic  or  exponential  curve,  log  y  =  a  +  bx  +  c*2 
or  y  =  aebx  +  cae\  —  These  equations  are  modifications  of  the  logarithmic 
form  log  y  =  a  +  bx  and  the  exponential  form  y  =  a&*.  The  equation 
y  =  aebl+c^  may  be  written  log  y  =  log  a  +  (b  log  e)  x  -\-  (c  log  e)  x2,  and 
so  is  equivalent  to  the  form  log  y  =  a  +  bx  +  cxz.  This  last  equation 
is  similar  in  form  to  the  equation  y  =  a  +  bx  +  ex2  discussed  in,  Art.  77, 
and  the  equation  may  be  verified  and  the  constants  determined  in  a 
similar  way. 

Hence,  if  a  set  of  data  may  be  approximately  represented  by  the  equation 


hg  y  ~ 


where 


log  y  =  a  +  bx  +  ex2,    then    (i)    the    plot    of   (x, 

\ 

(xk,  yk)  are  the  coordinates  of  a  point  on  the  experimental  curve,  will  approxi- 
mate a  straight  line,  or  (2)  the  plot  of  (x,  A  log  y)  ,  where  the  A  log  y  are  the 
differences  in  log  y  formed  for  equidistant  values  of  x,  will  approximate  a 
straight  line. 

Example.  The  following  table  gives  the  results  of  Winkelmann's 
experiments  on  the  rate  of  cooling  of  a  body  in  air  ;  0  is  the  excess  of  tem- 
perature of  the  body  over  the  temperature  of  its  surroundings,  t  seconds 
from  the  beginning  of  the  experiment. 


t 

0 

logo 

log  8  —  log  118.97 

log  8  -log  1  18.97 

ee 

A 

I 

0 

118.97 

2.07544 

0 

118.97 

O 

12.  1 

116.97 

2.06808 

—0.00736 

—  O.OOo6o8 

116.99 

—  O.O2 

25-8 

114.97 

2.06059 

—0.01485 

—  0.000576 

114.97 

0 

41-7 

112.97 

2.05296 

—0.02248 

—  0.000539 

112.90 

+0.07 

59-7 
82.0 

110.97 
108.97 

2.04520 
2.03731 

—0.03024 
—0.03813 

—  0.000507 
—  0.000465 

110.90 
108.90 

+0.07 
+0.07 

109.0 

106.97 

2.02926 

—0.04618 

—  0.000424 

107.15 

-0.18 

In  Fig.  79  we  have  plotted  the  points  (t,  6}.  According  to  Newton's 
law  of  cooling,  6  =  ae*1  or  log  0  =  a  +  bt,  and  so  we  have  also  plotted  the 
points  (/,  log  6} ;  this  last  plot  has  a  slight  curvature.  We  shall  therefore 
assume  the  law  in  the  form  log  e  =  a  +  bt  +  ct2.  To  verify  this,  we 
note  the  point  tt  =  o,  dk  =  118.97  on  the  experimental  curve,  and  plot 

the  points  (/,  — —}',    this    plot    approximates    a   straight 

line,  so  that  we  may  assume  — ~ —  =  b  +  ct.    We  use  the 

method  of  averages  to  determine  the  constants.     Dividing  the  data  into 
two  groups  of  three  sets  each  and  adding,  we  get 

-0.001723  =  36  +    79-6  c, 

-0.001396  =  36  +  250.7  c. 


152  EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 

.'.     c  =  0.000001911,     b  =  —0.000625. 

log  6—  log  118.97 

— 7^-        —  =  —0.000625  +  0.000001911  / 

or  log  e  =  2.07544  -  0.000625  t  +  0.600001911  P. 

We  now  compute  6  and  the  residuals  and  note  the  close  agreement 
between  the  observed  and  calculated  values. 


11R  0? 

s 

Q.  00  06 
Q  000% 

116.97 
114.97 

(e) 

112.97 
110.97 
108.97 
106.97 

C 

X, 

\ 

N 

4 

X 

•^ 

^ 

\ 

w 

'""*** 

\: 

s^ 

K 

^ 

cf 

^c 

^3 

~^^ 

^ 

ff>! 

^x 

^ 

^» 

^ 

""v 

*>t 

7^ 

^^ 

^ 

-^ 

<J 

^ 

'^v 

-^ 

1  , 

-  — 

-~~- 

I         70       20       30       40       50       60       70       80       90      100      110 

FIG.  79. 

(IV)  EQUATIONS   INVOLVING  FOUR  OR  MORE  CONSTANTS. 

80.  The  additional  terms  cedx  and  cxd.  —  It  is  sometimes  found  that 
a  simple  equation  will  represent  a  part  of  our  data  very  well  and  another 
part  not  at  all,  i.e.,  the  residuals  y0  —  yc  are  very  small  for  one  part  of 
our  data  and  quite  large  for  another  part.  Geometrically,  this  is 
equivalent  to  saying  that  the  plot  of  the  simple  equation  coincides 
approximately  only  with  a  part  of  the  experimental  curve.  In  such 
cases  a  modification  of  the  simple  equation  by  the  addition  of  one  or 
more  terms  will  often  cause  the  curves  to  fit  approximately  throughout. 
Such  terms  usually  have  the  form  ce?x  or  cd*,  and  added  to  our  simple 
equations  give  the  forms 

y  =  a  +  bx  +  ce?*,  y  =  a  +  bx  +  ex?, 

y  =  aef"  +  cef*.  y  =  ax*  +  cxd, 


y  = 


+  cxd,      etc. 


, 
a  +  bx  a  +  bx 

We  shall  give  a  few  examples  to  illustrate  some  of  these  cases. 


ART.  8 1 


THE  EQUATION 


+  bx  +  ce** 


'S3 


A 


8i.  The  equation  y  =  a  +  bx  +  ce*". —  If  a  part  of  the  experimental 
curve  approximates  a  straight  line,  we  may  fit  an  equation  of  the  form 
y  =  a  +  bx  to  this  part  of  the  curve.  The  deviation  of  this  straight 
line  from  the  remainder  of 
the  experimental  curve  (Fig. 
8 1  a)  will  be  measured  by  the 
residuals  r  =  y0  —  yc  =  y  — 
(a  +  bx).  We  now  plot  (x,  r) 
and  study  the  nature  of  this 
plot.  We  may  be  able  to 
represent  this  plot  by  means 
of  the  simple  exponential  r  = 
cedx,  where  the  values  of  the 
constants  c  and  d  are  such 
that  the  value  of  r  is  negli- 
gible for  that  part  of  the  plot 
to  which  the  straight  line  has 
been  fitted.  The  entire  ex- 
perimental curve  can  thus  be 
represented  by  ce?1  =  y  -  (a  FlG>  8/0/ 

+  bx}  or  y  =  a  +  bx  +  cedx. 

The  equation  y  =  a  +  bx  +  ce?x  may  fit  an  experimental  curve 
although  no  part  of  the  curve  is  approximately  a  straight  line;  this 
means  that  the  values  of  the  term  cedx  are  not  negligible  for  any  values 
of  x.  If  the  values  of  x  are  equidistant,  we  may  verify  that  this  equa- 
tion is  the  correct  one  to  assume  by  the  following  method.  Let  the 
constant  difference  in  the  values  of  x  be  h.  If  we  replace  x  by  x  +  h, 
we  get 


(y)s 

4 
3 
2 
1 


8      9     10 


y  =  a          * 
and,  therefore,  for  the  difference  in  the  values  of  y, 

Ay  =  y'  -  y  =  bh  +  cedx  (edh  -  i). 
If  Ay  and  Ay'  are  two  successive  values  of  Ay,  then 

Ay'  =  bh +  €*(*+»  (e*  -  i), 
and  the  difference  in  the  values  of  Ay  is 

A'y  =  Ay'  -  Ay  =  ct**  (edh  -  i)2. 

log  A2y  =  log  c  (€*»>  -  i)2  +  (d  log  e)  x. 


Hence, 


The  last  equation  is  of  the  first  degree  in  x  and  log  A2y  so  that  the  plot 
of  (x,  logA2y)  will  approximate  a  straight  line.     From  this  straight 


154 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


line  we  may  determine  the  constants  log  c  (edh  —  i)2  and  d  log  e  and 
therefore  c  and  d  in  the  usual  way.     We  now  write  the  equation  in  the 

form  y  —  cedx  =  a  -\-  bx,  and 
from  the  straight  line  plot  of 
(x,  y  —  cedx),  we  determine  the 
constants  a  and  b. 

In  Fig.  8ib  we  have  plotted 
the  equations 

y  =  0.5  +  x, 
y  =  0.5  +  x  —  o.oi  e1, 
y  =  0.5  +-  x  —  o.ooi  ex, 
y  =  0.5  +  x  +  o.oi  ex, 
y  =  0.5  +-  x  +  o.ooi  ex. 


Example.  The  following 
data  are  the  results  of  experi- 
ments made  with  a  gasometer 
by  means  of  which  the  amount 
of  air  which  passes  into  a  re- 
ceiving tank  can  be  measured; 
x  is  the  vacuum  in  the  tank  in 
inches  of  mercury,  y  is  the 
number  of  cu.  ft.  of  air  per 
minute  passing  into  the  tank. 
(Experiments  made  by  W.  D. 
Canan  at  the  Mass.  Inst.  of 
Tech.) 


FIG.  8 1  b. 


X 

V 

»' 

r  =  y1  —  y 

logr 

r. 

Vc 

A 

8 

•17 

•49 

0-32 

9.5051  —  10 

0.322 

•  17 

0 

10 

•37 

•55 

0.18 

9.2553  -  10 

0.179 

•  37 

0 

12 

•So 

.61 

O.II 

9.0414  —  10 

0.099 

.51 

—o.oi 

H 

.62 

.67 

0.05 

8.6990  —  10 

0-055 

.61 

+  0.01 

16 

•7i 

•73 

0.02 

0.031 

•70 

+O.OI 

18 

.80 

•79 

—  O.OI 

0.017 

•77 

+0.03  ' 

20 

•85 

•85 

0 

0.009 

.84 

+  0.01 

22 

•9i 

.91 

0 

0.005 

•90 

+0.01 

24 

.96 

•97 

O.OI 

0.003 

•97 

—  O.OI 

26 

.02 

•°3 

O.OI 

0.002 

•°3 

—o.oi 

28 

.  10 

.09 

—  o.oi 

O.OOI 

•09 

+O.OI 

In  Fig.  Sic  we  note  that  the  plot  of  (x,  y)  approximates  a  straight 
line  for  values  of  x  >  14,  and  we  shall  fit  an  equation  of  the  form 


ART.  8 1 


THE   EQUATION 


+  bx  +  ce** 


155 


y'  =  a  +  bx  to  this  part  of  the  data.     Using  the  method  of  averages 
and  dividing  the  data  into  two  groups  of  four  and  three  sets,  we  have 

7.27  =  40  +  76  b, 

6.08  =  3  a  +  78  b, 
.:     b  =  0.03,    a  =  1.25 
and  y'  =  1.25  +  0.03*. 


2.1 

2.0 

/ 

p.ff 

0.4 

5.2  -r 
S 

3.0  ~ 
«.ff 

/* 

r 

s 

/_ 

f 

1.9 
1.8 
1.7 

1.5 
1.4 
1.3 
1.2 
1.1 
1  n 

\ 

d 

S 

\ 

/ 

\ 

^ 

/ 

''^ 

S 

* 

y& 

^ 

f 

\ 

, 

'', 

/ 

fs* 

\  7 

A 

/ 

\ 

A 

j 

\ 

/ 

<*" 

7 

i  n 

\ 

y 

^ 

>  2 

1 

* 

\ 

1 

x 

YCt 

^ 

Jrl 

[ 

X 

<^ 

** 

^^^ 

•  —  , 

n 

S         /0        12        1.4 


16 


18       20 
(x) 

FIG.  8 ic. 


22       24       26       2S 


Now  compute  the  values  of  y'  and  the  residuals  r  =  y'  —  y  (by 

taking  r  =  y'  —  y  instead  of  r  =  y  —  y',  the  residuals  are  positive  and 

easier  to  handle  in  the  subsequent  calculations).     Plot  (x,  r}  for  values 

of  x  <  14  and  study  the  nature  of  this  plot;  this  seems  to  be  a  simple 

exponential,  r  —  cedx]  verify  this  by  plotting  (x,  log  r)  and  note  that 

this  plot  approximates  a  straight  line.     Using  the  method  of  averages 

determine  the  constants  in  the  equation  log  r  ='log  c  +  (d  log  e)  x;  thus 

8.7604  —  10  =  2  log  c  +  18  d  log  e, 

7.7404  —  10  =  2  log  c  +  26  d  log  e. 


156  EMPIRICAL  FORMULAS  — NON-PERIODIC  CURVES         CHAP.  VI 

/.     d  log  e  =  9.8725  -  10  =  -0.1275,     logc  =  0.5277. 

/.     d  =  -  0.294,     c  =  3-37- 

/.     log  r  =  0.5277  -  0.1275  x,     and    r  =  3.37  e-0-894* 

The  final  equation  is 

y  =  1.25  +  0.03  x  -  3.37  e"0-294*. 

Now  compute  y  and  the  residuals,  and  note  the  close  agreement  be- 
tween the  observed  and  calculated  values. 

82.  The  equation  y  =  aeba>  +  cedx.  —  A  part  of  the  experimental 
curve  may  be  represented  by  a  simple  exponential  y  =  ad3*,  i.e.,  a  part 
of  the  plot  of  (x,  log  y)  approximates  a  straight  line.  We  then  study 
the  deviations,  r  =  y0  —  yc  =  y  —  ae**,  of  this  exponential  curve  from 
the  rest  of  the  experimental  curve.  The  plot  of  (x,  r}  may  be  repre- 
sentable  by  another  exponential,  r  =  cedx,  where  the  values  of  r  are 
negligible  for  that  part  of  the  experimental  curve  to  which  y  =  aeP*  has 
been  fitted.  The  entire  curve  can  then  be  represented  by  the  equa- 
tion y  =  a&x  +  cedx. 

The  equation  y  =  aePx  -f-  cedx  may  fit  an  experimental  curve  although 
no  part  of  the  curve  can  be  approximated  by  the  simple  exponential 
y  =  a^x.  If  the  values  of  x  are  equidistant,  we  may  verify  that  this 
equation  is  the  correct  one  to  assume  by  the  following  method.  Let 
the  constant  difference  in  the  values  of  x  be  h.  Consider  three  succes- 
sive values  x,  x  +  h,  x  +  2  h  and  their  corresponding  values  y,  y',  y". 
We  evidently  have 

y  =  ae*>x  +  cedx, 

y'  =  ag*  (*+>>)  -f  £«*(*+*>  =  ae*x(*h  +  cedxedh, 
y"  =  ae6(*+2*)  +  <#*<*+«*>  =  a#*&™  +  cedlezdh. 

Now  eliminate  &x  and  edx  from  these  three  equations  by  multiplying 
the  first  equation  by  g(b+d)'1,  the  second  by  —  (e6*  +  edh),  and  adding  the 
results  to  the  third  equation.  We  get 

y"    _     (gW    +    gdh-)    y' 

or  y—  =  (ef'h  +  edh)  y- 

This  is  an  equation  of  the  first  degree  in  y'/y  and  y" /y  so  that  the  plot 
of  (y'/y,  y" /y)  will  approximate  a  straight  line.  From  this  straight  line 
determine  the  constants  e*h  +  c*  and  e^^,  and  hence  b  and  d  as  usual. 
We  now  write  the  original  equation  ye~dx  =  ae(b~d)x  +  c.  This  is  a  linear 
equation  in  gC^*  and  ye~dx  so  that  the  plot  of  (e^6-"")',  ye"*1}  would 
approximate  a  straight  line.  From  this  straight  line  determine  the 
values  of  the  constants  a  and  c. 


ART.  82 


THE  EQUATION  V  =  ae°*  +  «"* 


157 


In  Fig.  820,  we  have  plotted  the  equations  y  =  e~x,  y  =  e~x  +  0.5  «" 


t.O      .  x     1.5 
(x) 

y  =  a<*x  +  cedx 
FIG.  820,. 

Example.  The  following  are  the  measurements  made  on  a  curve 
recorded  by  an  oscillograph  representing  a  change  of  current  i  due  to  a 
change  in  the  conditions  of  an  electric  circuit  t.  (From  Steinmetz, 
Engineering  Mathematics.) 


1 

I 

logi 

»' 

r  =  t'  —  i 

logr 

re 

«c 

A 

0 

2.10 

O.3222 

4.94 

2.84 

0-4533 

2.85 

2.09 

-fo.oi 

O.I 

2.48 

0-3945 

4-44 

1.96 

0.2923 

1  .96 

2.48 

0 

0.2 
0.4 

2.66 
2.58 

0.4249 
0.4Il6 

3-99 
3-22 

1.33 
0.64 

0.1239 
9.8062  —  10 

1-34 
0.63 

2.65 

2-59 

+0.01 
—  O.OI 

0.8 

2.00 

O.30IO 

2.10 

O.IO 

9.0000  —  10 

0.14 

1.96 

+0.04 

I  .2 

1.36 

0-I33S 

i-37 

O.OI 

0.03 

1-34 

+O.O2 

1.6 

0.90 

9.9542  -  10 

0.89 

—  0.01 

O.OI 

0.88 

+O.O2 

2.0 

P.58 

9.7634  -  10 

0.58 

0 

o 

0.58 

O 

2-5 

0-34 

9.5315  -  10 

0-34 

o 

o 

0-34 

0 

3-0 

O.20 

9.3010  —  10 

0.20 

0 

o 

0.20 

O 

In  Fig.  826  we  note  that  the  right-hand  part  of  the  plot  of  (t,  i)  appears 
to  be  exponential.  We  verify  the  choice  of  i'  =  ad1  by  plotting  (/,  log  i) 
and  noting  that  this  plot  approximates  a  straight  line  for  values  of 


'58 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


i  >  0.8.     We  therefore  assume  log  i'  =  log  a  +  (b  log  e)  t,  and  using  the 
method  of  averages  for  the  values  of  t  >  0.8,  we  have 

9.8511  -  10  =  3  log  a  +  4.8  b  log  e, 
8.8325  -  10  =  2  log  a  +  5.5  b  log  e. 

0.6934, 


and 


bloge  =  9.5356  -  10  =  -0.4644,     log  a 
b  =  —  1.07,     a  =  4.94, 

\ogi'  =  0.6934  -  0.4644 /,     or     i'  =  4.94 erl-mt. 


5n 

n  s* 

\ 

N 

\ 

^ 

, 

^ 

\ 

0  4 

f 

7^ 

-^ 

ss 

\ 

5 

' 

\ 

X 

s\ 

0  3 

7 

\ 

'N 

XN 

/ 

\ 

N 

n  9 

/ 

\ 

\ 

\ 

9  n 

\ 

\ 

\ 

n  | 

] 

\ 

\ 

? 

\ 

% 

f, 

/)  /) 

1 

\ 

\ 

1  5 

g  ^ 

1 

, 

\ 

\ 

<£> 

9  9' 

\ 

\ 

s 

\ 

r? 

L 

V 

"N 

s  >>• 

\ 

^"*" 

q  o 

\ 

\ 

\ 

5 

1  0 

A- 

i 

\ 

\ 

\ 

97 

1 

\ 

\ 

\ 

\ 

\ 

\ 

g  /j 

\ 

\ 

\ 

s. 

1 

\ 

n  e 

y 

sTv 

\ 

^ 

X 

v 

Q    C 

/.t 

^) 

V 

\ 

"•s 

\, 

s 

0  ' 

\ 

~«. 

•--, 

^\ 

9f 

\ 

\ 

^ 

^-^ 

Q 

V 

^ 

\ 

,9,9 

C 

> 

0. 

5 

/. 

0 

1. 

(t 

5 
) 

2. 

0 

2. 

5 

3.( 

? 

Now  find  the  values  of  i'  and  the  residuals  r  =  i'  —  i;  these  residuals 
are  practically  negligible  for  values  of  t  >  0.8.  We  plot  (/,  r)  and  try 
to  fit  an  equation  to  this  curve.  This  again  appears  to  be  exponen- 
tial and  we  verify  this  by  plotting  (/,  log  r) ;  the  plot  approximates  a 


ART.  83  THE  POLYNOMIAL  y  =  a  +  bx  +  c*2  +  dx?  +  •  •  •  159 

straight  line,  except  for  t  =  0.8.    We  therefore  assume  r  =  cedi  or  log  r  = 
log  c  +  (d  log  e)  t.     Using  the  method  of  averages  for  t  <  0.8,  we  have 

0.7456  =  2  log  c  +  o.i  d  log  e, 
9.9301  —  10  =  2  log  c  +  0.6  d  log  e. 
.'.     dloge  =  -1.6310,     logc  =  0.4544. 
.'.  d  =  -3-76,     c  =  2.85, 

and    log  r  =  0.4544  -  1.6310  /,     or    r  =  2.85  g-376'. 
The  final  equation  is 

i  =  4.94  e~l-mt  —  2.85  e-3-™1. 

We  now  compute  i  and  the  residuals  and  note  the  very  close  agree- 
ment between  the  observed  and  computed  values  of  i. 

83.  The  polynomial  y  =  a  +  bx  +  ex*  -f-  dx3  +  •  •  •  .  — The  equa- 
tion y  =  a  +  bx  +  ex2  may  be  modified  by  the  addition  of  another 
term  into  y  =  a  +  bx  +  cxz  +  (fo3.     If  the  values  of  x  are  equidistant, 
we  may  verify  the  correctness  of  the  assumption  of  the  last  equation 
by  the  following  method.     Let  the  constant  difference  in  the  values  of 
x  be  h.     Then  the  successive  differences  in  the  values  of  y  are 
Ay  =  (bh  +  cW  +  dh3}  +  (2  ch  +  3  dtf)  x  +  3  dfcc», 
A2?  =  (2  cW  +  6  d&3)  +  6  d/*2x, 
A3;y  =  6  dh*. 

Hence  the  plot  of  (x,  A2;y)  will  approximate  a  straight  line,  and  the  values 
of  A33>  are  approximately  constant.  From  the  equation  of  the  straight 
line  we  may  determine  the  constants  c  and  d,  and  writing  the  original 
equation  in  the  form  (y  —  ex2  —  dx3)  =  a  +  bx,  the  plot  of  (x,  y—cx^  —  dx3) 
will  approximate  a  straight  line,  from  which  the  constants  a  and  b 
may  be  determined.  Another  method  of  determining  the  constants 
a,  b,  c,  d  in  the  equation  y  =  a  +  bx  +  ex*  +  dx?  consists  in  selecting 
four  points  on  the  experimental  curve,  substituting  their  coordinates 
in  the  equation,  and  solving  the  four  linear  equations  thus  obtained  for 
the  values  of  the  four  quantities  a,  b,  c,  and  d. 

In  a  similar  manner  the  polynomial  y  =  a  +  bx  +  cxz  -\-  •  •  •  +  kxn 
may  be  determined  so  that  the  corresponding  curve  passes  through 
n  +  i  points  of  the  experimental  curve;  it  is  simply  necessary  to  sub- 
stitute the  coordinates  of  these  n  +  i  points  in  the  equation  and  to  solve 
the  n  +  i  linear  equations  for  the  values  of  the  n  +  i  quantities,  a,  b, 
c,  .  .  .  ,  k.  If  the  values  of  x  are  equidistant,  we  can  show  that  the  plot 
of  (x,  An-1;y)  is  a  straight  line  and  that  An;y  is  constant,  where  An-1y 
and  A";y  are  the  (n  —  i)st  and  nth  order  of  differences  in  the  values  of  y. 
Thus,  if  a  sufficient  number  of  terms  are  taken  in  the  equation  of  the 
polynomial,  this  polynomial  may  be  made  to  represent  any  set  of  data 
exactly;  but  it  is  not  wise  to  force  a  fit  in  this  way,  since  the  deter- 
mination of  a  large  number  of  constants  is  very  laborious,  and  in  many 


i6o 


EMPIRICAL    FORMULAS  — NON-PERIODIC   CURVES         CHAP.  VI 


cases  a  much  simpler  equation  involving  fewer  constants  may  give  much 
more  accurate  results  in  subsequent  calculations. 

We  shall  work  a  single  example  to  illustrate  the  method  of  determin- 
ing the  constants. 

Example.  We  wish  to  fit  a  polynomial  equation  to  the  following 
data: 


&.0 
5.5 
5.0 
4.5 
4.0 
3.5 
3.0 

y) 

2.5 
2.0 
1.5 

n  20 

/ 

/ 

/I 

i 

0.19 
0.19 
0.14 
n  12 

j 

/ 

1 

^ 

/ 

\ 

i 

>/ 

/ 

/ 

/ 

? 

/ 

n  iff 

/ 

: 

« 

/ 

0.08 
0  06 

/ 

^ 

/ 

/ 

7 

n  04 

0.5 
n 

X 

X 

Q  Q9 

^ 

^ 

n 

0.1     0.2     0.3     0.4     0.5     0.6     0.7     0.8     0.9     1.0 
(X) 

FIG.  83. 


z 

V 

Av 

AV 

A'y 

Vc 

A 

o 

o 

O.2I2 

0.039 

O.OI9 

0 

0 

O.I 

O.2I2 

o  .251 

0.058 

O.OI4 

0.2IO 

+  O.002 

0.2 

.463 

0.309 

0.072 

0.09 

0.463 

O 

o-3 

•772 

0.381 

0.091 

o  .0  9 

0.770 

+  0.002 

0.4 

•153 

0.472 

O.IIO 

0.08 

I.I52 

-j-o.ooi 

°-5 

-625 

0.582 

0.128 

O.O    I 

1.625 

0 

0.6 

.207 

0.710 

0.149 

o.o  4 

2.209 

—  O.OO2 

0.7 

.917 

0.859 

0.163 

o.o  8 

2.92O 

—  0.003 

0.8 

3-776 

I  .022 

0.181 

3-776 

O 

0.9 

4.798 

I  .203 

4-797 

+O.OOI 

I.O 

6  .001 

S-998 

+0.003 

ART.  84 


TWO  OR  MORE  EQUATIONS 


161 


In  Fig.  83  we  have  plotted  (x,  y).  We  form  the  successive  differences 
and  note  that  the  third  differences  are  approximately  constant,  and 
that  the  plot  of  (x,  A2y)  approximates  a  straight  line  (Fig.  83).  We 
may  therefore  assume  an  equation  of  the  form  y  =  a  +  bx  +  cxz  +  dx3, 
or  y  =  bx  +  ex2  +  dx3,  since  the  curve  evidently  passes  through  the 
origin  of  coordinates.  To  determine  the  constants  b,  c,  and  d,  select 
three  points  on  the  experimental  curve;  three  such  points  are  (0.2,  0.463), 
(0.5,  1.625),  an<3  (0.8,  3.776).  Substituting  these  coordinates  in  the 
equation,  we  get 

0.463  =  0.2  b  +  0.04  c  +  0.008  d, 

i  .625  =  0.5  b  +  o.  25  c  +  0.125  d, 

3.776  =  0.8  b  +  0.64  c  +  0.512  d. 

Solving  these  equations  for  b,  c,  and  d,  we  have 

b  =  1.989,     c  =  1.037,     d  —  2.972 
and  hence  the  equation  is 

y  =  1 .989  x  +  1 .037  x*  +  2.972  x3. 

We  now  compute  the  values  of  y  and  the  residuals. 

84.  Two  or  more  equations.  —  It  is  sometimes  impossible  to  repre- 
sent a  set  of  data  by  a  simple  equation  involving  few  constants  or  even 
by  a  complex  equation  involving  many  constants.  In  such  cases  it  is 
often  convenient  to  represent  a  part  of  the  data  by  one  equation  and 
another  part  of  the  data  by  another  equation.  The  entire  set  of  data 
will  then  be  represented  by  two  equations,  each  equation  being  valid 
for  a  restricted  range  of  the  variables.  Thus,  Regnault  represented  the 
relation  between  the  vapor  pressure  and  the  temperature  of  water  by 
three  equations,  one  for  the  range  from  —  32°  F.  to  o°  F.,  another  for 
the  range  from  o°  F.  to  100°  F.,  and  a  third  for  the  range  from  100°  F. 
to  230°  F.  Later,  Rankine,  Marks,  and  others  represented  the  rela- 
tion by  a  single  equation.  The  following  example  will  illustrate  the 
representation  of  a  set  of  data  by  two  simple  equations. 

Example.  The  following  data  are  the  results  of  experiments  on  the 
collapsing  pressure,  p  in  pounds  per  sq.  in.  of  Bessemer  steel  lap-welded 
tubes,  where  d  is  the  outside  diameter  of  the  tube  in  inches  and  /  is  the 
thickness  of  the  wall  in  inches.  (Experiments  reported  by  R.  T.  Stew- 
art in  the  Trans.  Am.  Soc.  of  Mech.  Eng.,  Vol.  XXVII,  p.  730.) 


t 

d 

p 

*j 

logP 

p. 

A 

0.0165 

225 

8.2175  —  1° 

2.3522 

230 

—I 

0.0194 

383 

8.2878  -  10 

2.5832 

381 

+  2 

0.0216 

524 

8-3345  -  10 

2.7193 

533 

-9 

0.0214 

536 

8.3304  —  10 

2.7292 

Si? 

+19 

162 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


6000 
tfcnn 

8. 

(log  */&) 

2        8.3       S.4 

2.8 
2.7 

,st 

2.4 
2.3 

i 

/ 

i 

/ 

I 

y 

* 

^ 

/ 

KOQO 

/ 

/ 

A 

> 

/ 

4500 
4000 

/  j 

/ 

I 

/ 

j 

/ 

/ 

X 

3500 
(P) 
3000 

2560 
2000 
icnn 

f 

T 

/ 

/ 

/- 

+/ 

^ 

/  V 

-r 

^ 

\ 

J' 

i 

J 

1000 

? 

/ 

500 
n 

A" 

fi 

/ 

, 

^ 

T 

0.01     0.02     0.03     0.04     0.05     0.06     0.07      0.08     0.09    O.JO 

(Vd) 
FIG.  84. 


t 

d 

P 

Pe 

A 

* 
d 

P 

Pe 

A 

0.0228 

S92 

570 

+  22 

0.0370 

1779 

1821 

—  42 

0.0250 

670 

764 

-  84 

0.0374 

1860 

1856 

+  4 

0.0253 

870 

790 

+  80 

0-0375 

1879 

1865 

+  14 

0.0277 

928 

1002 

-  74 

o  .  0400 

2147 

2085 

+  62 

0.0298 

964 

Il87 

—  223 

0.0403 

2224 

2112 

+  112 

0.0299 

1184 

II96 

—  12 

o  .0436 

2280 

2403 

-123 

0.0309 

1251 

1284 

-  33 

o  .0442 

2441 

2455 

—  14 

0.0316 

1319 

1346 

—  27 

0.0477 

2962 

2764 

+  198 

0.0309 

1419 

1284 

+  135 

0.0527 

3170 

3204 

-  34 

0.0343 

1680 

1583 

+  97 

0.0628 

4095 

4194 

-  99 

0.0349 

1762 

1636 

+  126 

0.0815 

SS6o 

5741 

-181 

ABT.  84  TWO  OR  MORE  EQUATIONS  163 

It  should  be  noted  that  a  set  of  corresponding  values  of  t/d  and  P 
are  not  the  results  of  a  single  experiment  but  the  averages  of  groups 
containing  from  two  to  twenty  experiments. 

Following  the  work  of  Prof.  Stewart,  we  have  plotted  (t/d,  P),  Fig.  84, 
and  note  that  the  experimental  curve  approximates  a  straight  line  for 
all  values  of  t/d  except  the  first  four,  i.e.,  for  values  of  t/d  >  0.023. 

We  may  therefore  assume  P  =  a  +  bl-j).     If  we  use  the  method  of 

W 

selected  points  to  determine  the  constants  a  and  b  we  may  choose  the 
points  t/d  =  0.065,  P  =  4250,  and  t/d  =  0.030,  P  =  1215  as  lying  on 
the  straight  line;  we  then  have 

4250  =  a  +  0.065  6, 

1215  =  a  +  0.0306. 

.'.     b  =  86,714,     a  =  -1386 
and  P  =  86,714^)-  1386. 

This  result  agrees  with  that  given  by  Prof.  Stewart.     If  we  use  the 

method  of  averages  to  determine  the  constants  a  and  b  we  divide  the 

last  22  sets  of  data  into  two  groups  of  n  each,  and  get 

12,639  =  110  +  0.32316, 

30.397  =  1 1  a  +  0.5247  b. 

.:     b  =  88,085,    a  =  -1438, 

and  P  =  88,055  Q  -  H38. 

In  our  table  we  have  given  the  values  of  P  computed  from  this  last 
formula.  The  values  of  P  computed  from  the  first  formula  agree  very 
closely  with  these.  It  is  seen  that  the  percentage  deviations  are  in 
general  quite  small  though  large  in  a  few  cases,  varying  from  0.2  per 
cent  to  10  per  cent,  which  is  to  be  expected  from  the  nature  of  the 
experiments. 

We  now  attempt  to  fit  an  equation  to  the  first  four  sets  of  data.     The 

/A*  k- 

addition  of  a  modifying  term  of  the  formic  (-3)    or  ce  d  to  the  above 

formula  is  not  successful  here.  We  shall  therefore  follow  Prof.  Stew- 
art's work  and  attempt  to  fit  an  equation  of  the  parabolic  form,  P  = 

/A6  ft  \ 

af-J  .     We  verify  this  choice  by  plotting  Mog-^,  logP)  and  observing 

that  this  plot  approximates  a  straight  line.  (The  fewness  of  the  ex- 
periments for  values  of  t/d  <  0.023  is  a  handicap  here.)  Assuming 


164 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  V! 


and  using  the  method  of  averages,  we  find 

4.9354  =  2  log  a  +  (6.5053  -  10)  b, 
5.4485  =  2  log  a  +  (6.6649  -  10)  b. 

.:     b  =  3.11,    a  =  80,580,000 

/A3.11 

and  P  =  80,580,000  (-, 


We  compute  the  values  of  P  from  this  formula. 

The  entire  set  of  data  have  thus  been  represented  by  means  of  two 
simple  equations,  each  valid  for  a  restricted  range  of  the  variables.* 


EXERCISES. 

[Note.  The  exercises  which  follow  are  divided  into  two  sets.  The  type  of  equation 
that  will  approximately  represent  the  empirical  data  is  suggested  for  each  example  in 
the  first  set.  For  the  examples  in  the  second  set,  the  choice  of  a  suitable  equation  is  left 
to  the  student.] 

I .  Temperature  coefficient ;  r  is  the  resistance  of  a  coil  of  wire  in  ohms,  6  is  the  tem- 
perature of  the  coil  in  degrees  Centigrade,  [y  =  a  +  bx] 

r\     10.421      I      10.939  11.321      I      11.799     I     12.242          12.668 


10.50      |      29.49  42.70      |      60.01       |      75.51  91.05 

2.   Galvanometer  deflection;  D  is  the  deflection  in  mm.,  I  is  the  current  in  micro- 
amperes,    [y  =  a  +  bx] 


29.1 


D 

T~|  0.0493 


48.2 


72.7 


J2.O 

118 

140 

165 

•J54 

o-i97  j. 

0.234 

0.274 

199 
0.328 


0.0821    I    0.123 

3.   Volt-ampere  characteristic  of  118  volt  tungsten  lamp;  e  is  the  terminal  voltage,  « 
is  the  current.     \y  =  axb] 

e  I 


2 

4 

8 

16 

25 

32 

50 

64  I  loo 

125 

0.0245 

0.0370 

0.0570 
e 

0.0855 
1  ISO 

0.1125 
1  80 

0.1295 

200 

0.1715 
218 

O.2OOO  1  O.26O5 

0.2965 

t 

1  0.3295 

0-3635 

0.3865 

0.4070 

4.   Pressure- volume  of  saturated  steam;  v  is  the  volume  in  cu.  ft.  of  I  pound  of 
steam,  p  is  the  pressure  in  pounds  per  sq.  in.     [y  =  axb] 


26.43    |    22.40    [    19.08 


14.70    I    17.53    |    20.80 


16.2 


I4-04 

28.83 


33-71 


9-147 
45-49 


7-995 
52-52 


5.    Chemical  concentration  experiment;  x  is  the  concentration  of  hydrogen  ions,  y  is 
the  concentration  of  undissociated  hydrochloric  acid.     \y  =  axb] 


1.68 

1.22 

0.784 

1.32 

0.676 

0.216 

0.426 

"00747 


0.092 

0.047 

0.0096 

0.0049 

0.00098 

0.0085 

0.00315 

0.00036 

0.00014 

0.000018 

6.   Vibration  of  a  long  pendulum;  A  is  the  amplitude  in  inches,  /  is  the  time  since  it 
was  set  swinging.     \y 


0 

I 

2 

3 

4 

5 

6 

10 

4-97 

2-47 

1.22 

0.61 

0.30 

0.14 

*  Prof.  Peddle  in  "  The  Construction  of  Graphical  Charts  "  has  fitted  the  equation 
t/d  =  0.00274  ^P  +  o.ooooooooii  P2  to  Prof.  Stewart's  data. 


EXERCISES 


165 


7.  Newton's  law  of  cooling;  0  is  the  excess  of  the  temperature  of  the  body  over  the 
temperature  of  its  surroundings,  t  is  the  time  in  seconds  since  the  beginning  of  the  experi- 
ment. \y  -  »ebx] 


19.9 


345 


18.9 


19-30 
14.9 


28.80 


12.9 


53-75 


70-95 


8.9 


6.9 


8.    Barometric  pressure;  p  is  the  pressure  in  inches  of  mercury,  h  is  the  height  in  ft. 
above  sea  level,     [y  =  ae*>*\ 


886 


29 


2753 


4763 


27 


6942 


10,593 


9.    Electric  arc  of  length  4  mm. ;  V  is  the  potential  difference  in  volts,  *  is  the  current 
in  amperes.     \y  =  *  +  - J 


2-97 


345 


3-96 


4-97 


5-97 


6.97 


7-97 


V         67.7          65-0     |     63.0         61.0     |    58.25    |    56.25    |    55.10      54.30 
10.   Speed  of  a  vessel;  H.P.  is  the  horse  power  developed,  »  is  the  speed  in  knots. 


H.P. 


290 


560 


1144 


ii 


12 


1810 


2300 


1 1 .   Hydraulic  transmission ;  H.P.  is  the  horsepower  supplied  at  one  end  of  a  line  of 
pipes,  u  is  the  useful  power  delivered  at  the  other  end.       I  —  =  a.  +  bx2 


H.P. 


150 


96.5 


138 


172 


.250 
196 


300 


206 


12.   Magnetic  characteristic  of  iron;  H  is  the  number  of  gilberts  per  cm.,  a  measure 
of  the  field  intensity,  B  is  the  number  of  kilolines  per  sq.  cm.,  a  measure  of  the  flux 


density,     [y  = 


13.0 


14.0 


154 


16.3 


17.2 


40 


17.8 


60 


80 


18.5 


18.8 


13.   Focal  distance  of  a  lens;  p  is  the  distance  of  the  object,  p'  is  the  distance  of  its 
image. 


140 


22.50 1  23.20 1  23.80 1  24.60 1 26.20 


29.00 


14.   Pressure-volume  in  a  gas  engine;  p  is  the  pressure  in  pounds  per  sq.  in.,  v  is  the 
volume  in  cu.  ft.  per  pound.     \y  =  axb  +  c] 


53-8 


85.8 


113.2 


7-03 


5-85 


1.90 


3-50      |       2.50 

15.   Law  of  cooling;  0  is  the  temperature  of  a  vessel  of  cooling  water,  /  is  the  time  In 
minutes  since  the  beginning  of  observation,     [y  =  aebas  +  c] 

'-,— ,— i-^— ,- 

74-5 


0  I      92.0 


85.3 


79-5 


67.0 


60.5 


53-5 


45-0 


39-5 


i66 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES          CHAP.  VI 


1 6.   Straw-fibre  friction  at  150  pounds  pressure  according  to  Goss's  experiments;  y 
is  the  coefficient  of  friction  for  a  straw-fibre  driver  and  an  iron  driven  wheel,  x  is  the  slip, 


0-153 

0.179 

0.213 

"J 

0.271 

0.313 

0-359 

0.368 

0.381 

0.386 

0.405 

0.56 

o.58 

f 

0.61 
0.411 

0.78 
0-432 

0-99 
0.458 

1.  10 

0.463 

1.04 
0.465 

1.22 
0-47 

1.40 

L_ 

i-75 

1.94 

2.OO 

2.25 

2-33 

3-15 

2-79 

17.    Expansion  of  mercury  according  to  Regnault's  experiments;  j  is  the  coefficient 
of  expansion  between  o°  C.  and  t°  C.     \y  =  a  +  bx  +  c*2] 


7  |0.< 


150 


250 


300 


360 


.00018179  |o.oooi82i6  10.00018261  0.0001832310.00018403  0.00018500  0.00018641 

18.   Velocity  of  water  in   Mississippi   River;   v  is  the  velocity,  D  is  the  depth. 
[y  =  a  +  bx  +  ex2] 


0 

O.I 

0.2 

0.3      0.4 

0-5 

0.6 

0.7 

0.8 

0.9 

3-1950 

3.2299 

3-2532 

3.26II  |  3.2516 

3.2282 

3.1807 

3.1266 

3-0594 

2-9759 

19.   Solution  of  potassium  chromate;  s  is  the  weight  of  potassium  chromate  which  will 
dissolve  in  100  parts  by  weight  of  water  at  a  temperature  of  t°  C.     [log  y  =  a+bx+cxz] 


o 

IO 

27.4 

42.1 

61.5 

62.1 

66.3 

70.3  - 

20.    Load-elongation  of  annealed  high  carbon  steel  wire  of  diameter  0.0693  and  gage 
length  30  in.;  Wis  the  load  in  pounds,  E  is  the  elongation  in  inches,     [y 


50 

100  1   150  1  200  1  225  ]  250 

260 

280 

290 

300 

310 

0.0130 

0.0251  0.0387  (0.0520  0.0589  (0.0659 

0.0689 

0.0746 

0.0778 

0.0807 

0.0842 

33Q  340  350 

0.0916   I  0.0980   |  o.i i ii 

21.   Load-elongation  of  wire  of  Ex.  20  in  hard-drawn  condition;  W  is  the  load  in 
pounds,  E  is  the  elongation  in  inches,     [y  =  a  -f-  bx  +  cxd] 


W 


0.0280 


0.0562 


0.0849 


400  1  500 

600 

700 

800 

850 

900 

0.115010.1471 

0.1820 

0.2191 

0.2628 

0.2879 

0.3166 

0.6 

o-9 

1.2 

i-5 

1.8 

2.1 

2-4 

3-o 

1.27 

0.88 

0.63 

0.46 

o-33 

0-25 

0.18 

O.IO 

22.  Empirical  curve.     \y  =  ae*""  +  cedas] 

_JLl_°_J_M_ 
y    I  3.00    |  1.89 

23.  Magnetic  characteristic  of  iron;  H  is  the  number  of  gilberts  per  cm.,  a  measure 
of  the  field  intensity,  B  is  the  number  of  kilolines  per  sq.  cm.,  a  measure  of  the  flux  density 

(cf.  Ex.  12). 


3-o 


8.4 


6 

8 

10 

1  1.  2 

13.0 

14.0 

15 

15-4 


20 

30 

40 

60 

80 

16.3 

17.2 

I7.8 

18.5 

18.8 

24.   Speed  of  a  vessel;  /  is  the  indicated  horsepower,  v  is  the  speed  in  knots. 
[y  =  a  +  bx  +  ex1  +  dx-] 


8 

9 

10 

II 

12 

13 

14 

15 

16 

17 

18 

1000 

1400 

1900 

2500 

3250 

4200 

5400 

6950 

8950 

11,450 

15,400 

EXERCISES 


I67 


25.   Test  on  square  steel  wire  for  winding  guns;  5  is  the  stress  in  pounds  per  sq.  in., 
£  is  the  elongation  in  inches  per  inch. 

S      5000  I  10,000     20,000     30,000     40,000      50,000     60,000      70.000      80,000 


E 


-I- 
|o.c 


OOOI9  O.OOO57  O.OOO94  O.OOI34  O.OOI73  O.OO2I6  O.OO256  O.OO297j 


0.00343 


100,000    110,000 

0.00390  (0.00444 


26.   Flow  of  water  over  a  Thomson  gauge  notch;  Q  is  the  number  of  cu.  ft.  of  water, 
H  is  the  head  in  feet. 


4.2 


6.1 


1.6 


8-5 


n-5 


2.4 


14.9 


23-5 


27.  Friction  between  belt  and  pulley;  9  is  the  arc  of  contact  in  radians  between  belt 
and  pulley,  P  is  the  pull  in  pounds  applied  to  one  end  of  pulley  to  raise  a  weight  W  at  the 
other  end.  ".- 

HIT 
6 


5.62          6.93  8.52          10.50        12.90         15.96         19.67        24.24        29.94 

28.   Electric  arc  of  length  2  mm.;  V  is  the  potential  difference  in  volts,  i  is  the  cur- 
rent in  amperes. 
•  I    1.96     I    2.46  I    2.97     I    3.45     I    3.96     I    4.97         5.97     I    6.97         7.97        9.00 


V\  50.25   I  48.70]  47.90  I  47.50  I  46.80   I  45.70     45.00   I  44.00     43.60      43.50 

29.    Normal  induction  curve  for  transformer  steel;  H  is  the  number  of  gilberts  per 
cm.,  B  is  the  number  of  lines  per  sq.  cm. 

H\ 


_     I.O      1.3        2.1' 

B  |  425  I  800  I  1750 


2.9 


2850 


34 


4.1 


4300  I  6100 


4-5 


6725 


5-2         5-9 


7800  |  8600  Jio, 


7-5    I    9-Q    I 

0,20o|lI,I50| 


II.O 


12,200 


30.    Pressure- volume  in  a  gas  engine;  p  is  the  pressure  in  pounds  per  sq.  in.,  v  is  the 
volume  in  cu.  ft.  per  pound. 


i 

39-6 

44-7 

53-8 

V 

10.61 

9-73 

8-55 

85-8 

113-2 

135-8 

178.2 

6.23 

5-18 

4-59 

3-87 

31.   Melting  point  of  alloy  of  lead  and  zinc;  6  is  the  temperature  in  degrees  Centi- 
grade, x  is  %  of  lead. 

40  50        I        60       I        70       I        80        I        90 


1 86 
32.    Empirical  curve. 


205 


226 


250 


276, 


304 


5 

7 

9 

II 

13 

11.03 

14.03 

17-53 

21-55 

26.12 

6.42      I     8 

33.   Candle-power  of  an  incandescent  lamp;  H  is  the  age  of  the  lamp  in  hours,  C.P. 
is  the  candle-power. 


H 


C.P. 


24.0 


17-6 


50Q 
16.5 


750 

1000 

1250 

15.8 

15.3 

14.9 

34.    Insulation  resistance-current  passes  through  insulator  and  galvanometer;  D  is 
the  deflection  of  the  galvanometer,  /  is  the  time  in  minutes. 


8.0 


6.2 


6    I    7 

... 

5.0  |  4.4 


4.0 


9 

3-5     3-3 


II        12        13 
3.0   |  "27"  |  2.5 


JiJ-JS- 
2.5  |  2.4 


i68 


EMPIRICAL  FORMULAS  —  NON-PERIODIC   CURVES         CHAP.  VI 


35.    Experiments  with  a  crane;  /  is  the  force  in  pounds  which  will  just  overcome  a 
weight  w. 


8-5 


12.8 


300 


400 


500 


25-6 


600 


I      70Q 
I     34-2 


800 


36.   Copper-nickel  thermocouple;  t  is  the  temperature  in  degrees,  p  is  the  thermo- 
electric power  in  microvolts. 


24 


37.   Law  of  falling  body;  5  is  the  distance  in  cm.  fallen  by  body  in  /  sec. 


38.  Loads,  which  cause  the  failure  of  long  wrought-iron  columns  with  rounded  ends; 
P/a  is  the  load  in  pounds  per  sq.  in.,  l/r  is  the  ratio  of  length  of  column  to  the  least 
radius  of  gyration  of  its  cross-section. 

l/r  140  1 80  220  260 


300 


340 


380 


420 


P/a      \    12,800        7500          5000          3800          2800     |     2100     I     1700          1300 

39.   Heat  conduction  of  asbestos;  0  is  the  temperature  in  degrees  Fahrenheit,  C  is 
the  coefficient  of  conductivity. 

0  32 212 | 392  572  752  1 1 12 


1.048  1.346      |     1.451  1.499  1.548  1.644 

40.    Rubber-covered  wires  exposed  to  high  external  temperatures;  C  is  the  maximum 
current  in  amperes,  A  is  the  area  of  cross-section  in  sq.  in. 

3.2  5.9  9.0  22.0          42.0         68.0          84.0      I    102.0 


0.001810  I  0.004072  I  0.007052     0.02227     0.05000    0.09442     0.1250    I   0.1595 

41.    Pressure-volume  relation  for  an  air  compressor;  p  is  the  pressure,  v  is  the  volume. 
p     |        18  21       |      26.5      |      33.5      |        44  62 


0-635          0.556     |     0.475     I     0.397     I     0.321          0.243 

42.    Power  delivered  by  an  electric  station;  w  is  the  average  weight  of  coal  consumed 
per  hour  per  kilowatt  delivered,  /  is  the  load  factor. 


f 

0.25 

0.20 

0.15 

0.10 

w 

2.843 

3-012 

3-293 

3-856 

43.   Temperature  at  different  depths  in  an  artesian  well;  0  is  the  temperature  in 
degrees  C.,  d  is  the  depth. 

d  28       |       66  173      I      248      I      298  400      I      505  548 


11.71 


12.90 


16.40 


23.75    I     26.45 


27.70 


44.    Resistance  of  copper  wire;  R  is  the  resistance  in  ohms  per  1000  ft.,  D  is  the 
diameter  of  wire  in  mils. 
D 


R 


289 


182 


102 


57 


iS 


10 


3.234         10.26    |     32.80    |     105.1 

45.  Hysteresis  losses  in  soft  sheet  iron  subjected  to  an  alternating  magnetic  flux; 
B  is  the  flux  density  in  kilolines  per  sq.  in.,  P  is  the  number  of  watts  lost  per  cu.  in.  for 
I  cycle  per  sec. 


40 


0.0022     I     0.0067 


60 


So 


100 


0.0128     j     0.0202     I     0.0289          0.0387 


120 


EXERCISES 


I69 


46.  Volt-ampere  characteristic  of  a  60  watt  tungsten  lamp:  V  is  the  number  of 
volts,  /  is  the  number  of  milli-amperes. 


V 

2 

5 

10 

20 

3° 

40 

50 

60 

70  1  80 

90 

100 

I 

V 

49 
no 

80 

120 

117 
130 

1  80 
140 

227 
1150 

272 
160 

3ii 
170 

348 
1  80 

383  1  4H 

190  1  200 

443 

210 

473 
220 

I 

501 

526 

553 

577 

597 

618 

639 

663 

682  1  702 

722 

743 

47.   Calibration  of  base  metal  pyrometer  (40%  Ni  and  60%  Cu);  V  is  the  number 
of  millivolts,  /  is  the  temperature  in  degrees  F. 

10      I      12      I      14      I      16 


t      |      o  146         255         320    |    396         475    |    553    |    634    |    714 

48.  Tests  on  drying  of  twine;  /  is  the  drying  time  in  minutes  (time  of  contact  of 
twine  with  hot  drum),  W  is  the  percentage  of  total  water  on  bone  dry  twine  at  any 
time,  E  is  the  percentage  of  total  water  on  bone  dry  twine  at  equilibrium,  d  is  the  di- 
ameter of  the  twine  in  ins. 

(a)  d  =  0.102  ins.,  E  =  18.7%. 


t 

o 

0-44 

0.88 

i-3i 

i-75 

W-E 

29-5 

15-4 

9-4 

5-1 

3-i 

6.2%. 


1 

0 

i.  ii 

2.23 

3-34 

4-45      1     5-56 

W-E 

30.3 

17.4 

12.4 

8.2 

4-9       1      3-3 

CHAPTER  VII. 
EMPIRICAL  FORMULAS  —  PERIODIC   CURVES. 

85.  Representation  of  periodic  phenomena.  —  Periodic  phenomena, 
such  as  alternating  electric  currents  and  alternating  voltages,  valve-gear 
motions,  propagation  of  sound  waves,  heat  waves,  tidal  observations, 
etc.,  may  be  represented  graphically  by  curves  composed  of  a  repetition 
of  congruent  parts  at  certain  intervals.     Such  a  periodic  curve  may  in 
turn  be  represented  analytically  by  a  periodic  function  of  a  variable, 
i.e.,  by  a  function  such  ihatf(x  -f-  k)  =  f(x),  where  k  is  the  period.     Thus 
the  functions  sin  x  and  cos  x  have  a  period  2  TT,  since  sin  (x  +  2  TT)  =  sin  x 
and  cos  (x  -\-  2  TT)  =  cos  x.     Again,   the  function  sin  5  x  has  a  period 
2  7T/5,  since  sin  5  (x  +  2  7r/5)  =  sin  (5  x  +  2  TT)  =  sin  5  x,  but  the  func- 
tion sin  x  -\-  sin  5  x  has  a  period  2  TT,  since  sin  (x  -f  2  TT)  +  sin  5  (x  +  2  TT) 
=  sin  x  +  sin  5  x. 

Now,  any  single-valued  periodic  function  can,  in  general,  be  expressed 
by  an  infinite  trigonometric  series  or  Fourier's  series  of  the  form 

y  =  f(x)  =  a0  +  a\  cos  x  +  0,1  cos  2  x  +  •  •  •   +  an  cos  nx  -\-  •  •  • 
+  61  sin  x  +  bz  sin  2  x  -f-  •  •  •   -}-  bnsmnx  -\-  •  -  -  , 

where  the  coefficients  a^  and  bk  may  be  determined  if  the  function  is 
known.  This  series  has  a  period  2  TT.  But  usually  the  function  is  un- 
known. Thus,  in  the  problems  mentioned  above,  the  curve  may  either 
be  drawn  by  an  oscillograph  or  by  other  instruments,  or  the  values  of  the 
ordinates  may  be  given  by  means  of  which  the  curve  may  be  drawn. 
Our  problem  then  is  to  represent  this  curve  approximately  by  a  series  of 
the  above  form,  containing  a  finite  number  of  terms,  and  to  find  the 
approximate  values  of  the  coefficients  ak  and  bk.  The  following  sections 
will  give  some  of  the  methods  employed  to  determine  these  coefficients. 

86.  The  fundamental  and  the  harmonics  of  a  trigonometric  series.  — 
In  Fig.  86a  we  have  drawn  the  curves  y  =  av  cos  x,  y  =  bi  sin  x,  and 
y  —  a\  cos  x  -f  b\  sin  x. 

The  maximum  height  or  amplitude  oi  y  =  a\  cos  x  is  fli  and  the 
period  is  2  TT.  The  amplitude  of  y  =  bi  sin  x  is  bi  and  the  period  is  2  TT. 
Now  we  may  write 


170 


ART.  86 


TRIGONOMETRIC   SERIES 


171 


and  letting     Vc 

we  may  write 

y  =  Ci  sin  (x  + 


=  cos  <£i, 


where 


Here  Ci  is  the  amplitude  and  $1  is  called  the  phase.  The  wave  rep- 
resented by  y  =  Ci  sin  (x  +  <£i)  is  called  the  fundamental  wave  and 
y  =  di  cos  x,  y  =  bi  sin  x  are  called  its  components. 


FIG.  86a. 

Similarly,  we  may  represent  y  =  ak  cos  jfoc,  y  =  bk  sin  fcc, 
and  y  =  a*  cos  kx  +  6*  sin  kx  =  ck  sin  (Jtoc  +  </>*), 

where  c*  =  Vak2  +  ft*2  and  0*  =  tan"1  ak/bk. 

The  wave  represented  by  y  —  ck  sin  (kx  -f-  <#>*)  is  called  the  kth  har- 
monic, its  amplitude  is  ck,  its  phase  is  fa,  its  period  is  2  T/£,  since 


in  Iktx  +  ^  +  0J  =  sin  [k 


x  +  2  T  +  **]  =  sin  (feac  +  0fc), 
and  its  frequency,  or  the  number  of  complete  waves  in  the  interval  2  T, 

13  k. 

The  trigonometric  series  is  often  written  in  the  form 
y  =  c0  +ci  sin  Ot+<£i)  +c2  sin  (2x-f<£2)  +  •  •  •  +  cn  sin  (w#  +  <£n)  +  •  •  •  , 
showing  explicitly  the  expressions  for  the  fundamental  wave  and  the 
successive  harmonics.  The  more  complex  wave  represented  by  this 
expression  may  be  built  up  by  a  combination  of  the  waves  represented  by 
the  various  harmonics.  Fig.  866  shows  how  the  wave  for  the  equation 


y  =  cos* 


•v/  -i 

--  - 


*\/  o 


cos  2*4 cos  3  x-]-  V3  sin  x  —  ?  sin  2  x  — 


sn  3  x 


172 


EMPIRICAL  FORMULAS  —  PERIODIC  CURVES 


CHAP.  VII 


is  built  up  as  the  combination  of  the  fundamental  and  the  second  and 
third  harmonics,  and  how  the  fundamental  wave  is  modified  by  the  addi- 
tion of  the  harmonic  waves. 


FIG.  86&. 

In  the  case  of  alternating  currents  or  voltages,  the  portion  of  the  wave 
extending  from  #  =  7rto:x;  =  27ris  merely  a  repetition  below  the  #-axis 
of  the  portion  of  the  wave  extending  from  x  =  o  to  x  =  TT;  this  is  illus- 
trated in  Fig.  S6c  where  the  values  of 
the  ordinate  at  x  =  xr  +  TT  is  minus 
the  value  of  the  ordinate  at  x  =  xr. 
/  _  Since 

x    sin  (k  [x  +  TT]  +  00 

=  sin  (kx  -+-  0fc  -J-  kir) 
=  +sin  (kx  -f-  00  if  k  is  even 

FIG-  86c-  =  -sin  (kx  +  00  if  k  is  odd, 

the  series  can  contain  only  the  odd  harmonics  and  has  the  form 

y  =  c0  +  ci  sin  (x  +  00  +  Cz  sin  ($x  +  03)  +  d,  sin  (5  x  -f  06)  +  •  •  •  , 
or 

y  =  a0  +  a\  cos  x  +  as  cos  3  x  +  05  cos  5  ac  +  •  •  • 
+  bi  sin  x  +  63  sin  3  x  +  65  sin  5  x  +  •  •  •  • 


ART.  87 


DETERMINATION  OF  THE   CONSTANTS 


173 


87.  Determination  of  the  constants  when  the  function  is  known.  — 
Tf,  in  the  series 

y  =  f(X~)    =  a0  +  di  COS  X  +  02  COS  2  X  +    •    •    •    +  dn  COS  UX  +    •   •   • 

+  bi  sin  x  +  bz  sin  2  x  +  •  •  •  +  bn  sin  nx  -\-  •  •  •  , 
we  multiply  both  sides  by  dx  and  integrate  between  the  limits  o  and  2  *, 
we  have 

/     y  dx  =  a0  I     dx  +  a\  I      cos  x  dx  +  .  .  .  +  an  I      cos  nx  dx  +  •  •  • 
+  61   /      sin ac dx  +  .  .  .  +  bn  I      sinnxdx+  •  •  • 


sin  x 


cosx 


sinn* 


cosnx 


o  n 

=  2  irflo,  since  all  the  other  terms  vanish. 

If  we  multiply  both  sides  by  cos  kx  dx  and  integrate  between  the 
limits  o  and  2  TT,  we  have 

r»  /*2x  /»2x 

y  cos  kx  dx  =  a0  I      cos  fcc  dx  +  •  •  •  +  #*  I      cos2  fcc  <fcc  +  •  •  • 
Jo  Jo 

+  an  /      cos  wx  cos  kx  dx  +  •  •  •      +  bn  I      sin  nx  cos  fcc  J*  +  •  •  • 
Jo  Jo 


sinfcc 


sin  2  kx 


2k 


a 


sin  (n  —  k]  x      sin  (n  +  k)  x 


bn  I  cos  (w  —  k)  x      cos  (w  +  k)  x 
~  ~2~ \       n-k  n  +  k 

=  irak,  since  all  the  other  terms  vanish. 

Similarly,  if  we  multiply  both  sides  by  sin  kx  dx  and  integrate  be- 
tween the  limits  o  and  2  TT,  we  have 

r*  rzr  (*2, 

ysinkxdx  =  a0l     sinkxdx+  •  •  •  +an  I     cosnx  sin  kxdx+  •  •  • 


Jh  i      sin2  kx  dx  + 
'o 


+  &, 


sin  nx  sin  £x  d# 


0,) 

& 

cc 

coskx 
s(k-- 

0 

0 

x   ,  cos(k 

.    bk 

x 

27T 

0 
0 

sin  2  &ac 

+  «) 

il 

fc-« 

sin  (n  —  k} 

k 
x      sin  (n 

+  »* 

n  —  k                 n  -\-  k 

,  since  all  the  other  terms  vanish. 


174  EMPIRICAL   FORMULAS  — PERIODIC   CURVES  CHAP.  VII 

Collecting  our  results,  we  have 

I     C2*  i    C2'  I    T2r 

do  =  —  I      y  dx,     a*  =  -  I      y  cos  kx  dx,    ft*  =  -   I      y  sin  £x  dx, 

2  IT  Jo  IT  Jo  IT  Jo 

where  k  =  1,2,  3,  ....  Each  coefficient  may  thus  be  independently 
determined  and  thus  each  individual  harmonic  can  be  calculated  without 
calculating  the  preceding  harmonics. 

88.  Determination  of  the  constants  when  the  function  is  unknown.  — 
In  our  problems  the  function  is  unknown,  and  the  periodic  curve  is  drawn 
mechanically  or  a  set  of  ordinates  are  given  by  means  of  which  the  curve 
may  be  approximately  drawn.  We  shall  represent  the  curve  by  a  trig- 
onometric series  with  a  finite  number  of  terms.  We  divide  the  interval 
from  x  =  o  to  x  =  2  IT  into  n  equal  intervals  and  measure  the  first  n 
ordinates;  these  are  represented  by  the  table 


2  7T 

4  TT 

67T 

2  T 

,                  ,   2  7T 

0 

r  — 

X 

n 

n 

n 

n 

n 

X0 

Xi 

xz 

X3 

.  .  . 

Xr 

Xn-l 

y 

y0 

yi 

y* 

ys 

yr 

yn-i 

We  wish  to  determine  the  constants  in  the  equation 


y  =  aQ  +  ai  cos  x  +  • 
+  bi  sin  x  +  • 


+  a-K  cos  kx 

+  bk  sin  kx  + 


where  the  number  of  terms  is  n,  so  that  the  corresponding  curve  will  pass 
through  the  n  points  given  in  the  table.  Substituting  the  n  sets  of  values 
of  x  and  y  in  this  equation,  we  get  n  linear  equations  in  the  a's  and  Z>'s  of 
the  form 


3V  =  0o  +  a>\  cos  xr  + 
+  bi  sin  xr  + 


+  fffc  COS  kxr  + 

+  bk  sin  kxr  -\- 


where  r  takes  in  succession  the  values  o,  I,  2,  .  .  .  ,  n  —  I.     We  may 
now  solve  these  n  equations  for  the  a's  and  &'s. 

We  shall  first  state  two  theorems  in  Trigonometry  concerning  the 
sum  of  the  cosines  or  sines  of  n  angles  which  are  in  arithmetic  progression, 
viz.: 

Tcos  (a  +  f/3)  =  cos  a  +  cos  (a  +  )3)  +  cos  (a  +  2  0)  +  •  •  • 


sin  - 

+  cos  (a  +  [n  -  i]  |8)  = —  cos 

sin 


AKT.  88  DETERMINATION  OF  THE   CONSTANTS 

sin  (a  +  rj3)  =  sin  a  +  sin  (a  +  /3)  +  sin  (a  +  2  /3)  +  • 


175 


sm  — - 

2       . 


an 


If  we  let  a  =  o  and  8  =  1  — ,  these  become 
n 

2  IT      sin  lir        I  (n  —  i)  TT 


2,  2 TT       sin  iir         L  \n  —  i »  TT  .  .     , 

cos  rl  —  =  ;-  cos  — —  =  O,  since  sin  ITT  =  o, 
n         .    lir                 n 


sm  — 
n 


2,  2?r      sm  lir    .    I  (n  —  l)  IT  .          .    . 

sin  rl  —  =  r-  sin  — —  =  o,  since  sin  lir  =  o. 
n         .    lir                n 


*  We  may  prove  these  theorems  as  follows: 

By  means  of  the  well-known  trigonometric  identities 

2  cos  u  sin  v  =  sin  (u  -\-  v)  —  sin  (u  —  v),       2  sin  u  sin  v  =  cos  («  —  »)—  cos  (u  +  v) 
we  may  write  the  identities 


8  I         8\  I         p\ 

2  cos  a  sin  -  =  sm  I  a  +  -  1  —  sin  I  a  --  1  • 

2  cos  (a  +5)  sin  |  =  sin  f  a+^?)-sinf  a  +  |  V 


2cos(a+[»-i]/3)sin     =  sin 


Adding,  we  get 


sin{  a+  ~~^  —  / 


sin^ 

-W.fS.!, 


Adding,  we  get 
2  sin  |  ^sin  (o+r/3)  =cos  fa  -  |  J 


=  2  sin  fa  +  -— -  /3j  sii 


.    w/3 
sm  — 

2     . 
-r  sm 


176  EMPIRICAL  FORMULAS  —  PERIODIC  CURVES  CHAP.  VII 

for  all  values  of  /  except 

/  =  o,  when   ^cos  rl  —  =  5}  cos  o  =  n, 

**  n        ** 


I  =  n,  when   5}  cos  ^  —  =  5} 

*** 


cos  zrir  =  n. 


Since  xr  =  r  —  ,  we  may  finally  state  that 


o,  except  when  /  =  o  or  /  =  n 
=  n,  when  /  =  o  or  /  =  n. 
^sin  lxr  =  o  for  all  values  of  /. 

To  determine  a0  we  merely  add  the  n  equations,  and  get 

5j;yr  =  na0  +  •  •  •   +  ak  ]£)cos  kxr  +  •  •  •   +  ak  J)sin  kxr  +  •  •  • 
=  na0,  since  all  the  other  terms  vanish. 

To  determine  ak  we  multiply  each  of  the  n  equations  by  the  coefficient 
of  ak  in  that  equation,  i.e.,  by  cos  kxr,  and  add  the  n  resulting  equations; 
we  get 

2)yr  cos  kxr  =  a0  2JCOS  kxr  +  -  •  •  +  a*  2}  cos2  kxr  +  •  •  • 

~\~ap  2)  COS  pXr  COS  kxr  +    '    '    '     +  6p^sln  pXr  COS  kxr  +  •  •  •    . 

Now, 

]vcos  kxr  =  o; 

cos  kxr*  =  %  J)cos  (P  +  *)  Xr  +  ^  2}cos  ^  ~  ^^r  =  0; 
cos  fe^r*  =  ^  ^sin  (^?  +  k)  xr  +  |  ^sin  (p  —  k)  xr  =  o; 


.,  ,       n 
=  n,  if  k  =  — 

2 


*  We  use  the  trigonometric  identities 

2  cos  a  cos  ti  =  cos  («  +  u)  +  cos  (M  —  t>). 
2  sin  «  cos  v  =  sin  («  +  v )  +  sin  («  —  i»). 
2  sin  u  sin  »  =  cos  («  —  ^)  —  cos  («  +  »). 


ART.  88 
Hence, 


DETERMINATION  OF  THE   CONSTANTS 


r  cos  kxr  =  -  ak,  except  when  k  —  - 


i        t 
,  when  k  =  -• 


177 


To  determine  bk  we  multiply  each  of  the  n  equations  by  the  coefficient 
of  bk  in  that  equation,  i.e.,  by  sin  kxr,  and  add  the  n  resulting  equations; 
we  get 


r  sin  kxr 


in  kxr 


Now, 


+  bk 


sin  kxr 


r  sin  kxr 


r  sin  kxr*  = 
r  sin  kxr*  = 


p)  xr 


-  (i  —  cos  2  fccr)  =  -  -- 

2  2,        2. 


Hence,  ^r  sm  ^•!Cr  "  ~~  ^ 

Collecting  our  results,  we  have  finally 

°o  =  -  2yr  =  -  (yo  +  y\ 


l  (y0  - 


"2 

-  (y0 


O 

r  sin  ^»P  =  -  (y6  sin  ^ 


(k  —  p}  xr  =*  o; 
s  (^  f  ^)  *r  -  o; 


=  -  .  if 

2 


=  o,  if  k  =    . 


+  yn-i  cos  kxn-J  , 
+  ^»-i  sin  kxn  .L). 


*  We  use  the  trigonometric  identities 

2  cos  «  cos  »  =  cos  (M  +  f  )  +  cos  (w  —  »). 
2  sin  M  cos  v  =  sin  (M  +  »)  +  sin  (M  —  v). 
2  sin  «  sin  t  -  cos  (u  —  v)  —  cos  («  +  p). 


178  EMPIRICAL   FORMULAS  — PERIODIC   CURVES  CHAP.  VII 

If  n  is  an  even  integer,  -our  periodic  curve  is  now  represented  by  the 
equation 

y  =  a0  +  ai  cos  x  +  •  •  •   +  ak  cos  kx  +  •  •  •  +  an  cos  -  x 


x  +  •  •  •   +  bk  sin  fcc  -f-  •  •  •   +  &„     sin  I i )  x. 

2~l        \2         / 

The  n  coefficients  are  determined  as  above.     Thus  — 

a0  is  the  average  value  of  the  n  ordinates. 

an  is  the  average  value  of  the  n  ordinates  taken  alternately  plus  and 


ak  or  bk  is  twice  the  average  value  of  the  products  formed  by  multiply- 
ing each  ordinate  by  the  cosine  or  sine  of  k  times  the  corresponding  value 
of*.* 

We  note  that  each  coefficient  is  determined  independently  of  all  the 
others. 

If  we  wished  to  represent  the  periodic  curve  by  a  Fourier's  series  con- 
taining n  terms,  but  had  measured  m  ordinates,  where  m  >  n,  we  should 
have  to  determine  the  coefficients  by  the  method  of  least  squares.  The 
values  of  the  ordinates  as  computed  from  this  series  will  agree  as  closely 
as  possible  with  the  values  of  the  measured  ordinates.  It  may  be  shown 
that  the  expressions  for  the  coefficients  obtained  by  the  method  of  least 
squares  have  the  same  form  as  those  derived  above. f 

*  We  may  also  derive  the  values  of  the  coefficients  as  follows:  In  Art.  87,  we  have 
shown  that 

J    *  y  cos  kx  dx  =  ak  J    *  cos2  kx  dx, 

since  all  the  other  terms  vanish. 

If  we  replace  the  integrals  by  sums,  and  take  for  dx  the  interval  2  v/n,  this  becomes 

>.yr  cos  kxr  =  ak  ^cos2  kxr  =  —ak,  if  k  ^  o  or  k  ^ 
=  nak,  if  k  =  o  or  k  = 

Hence,  Vyr  =  noo,     Vyr  cos  -  xr  =  nan,     ^yr  cos  kxr  = 

Similarly  we  may  show  that  ^yr  sin  kxr  =  —  bk. 

t  See  A  Course  in  Fourier's  Analysis  and  Periodogram  Analysis  by  G.  A.  Carse  and 
G.  Shearer. 


ART.  89 


NUMERICAL  EVALUATION  OF  THE  COEFFICIENTS 


179 


We  shall  illustrate  the  use  of  the  above  formulas  for  the  coefficients  by 
finding  the  fifth  harmonic  in  the  equation  of  the  periodic  curve  passing 
through  the  12  points  given  by  the  following  data  (Fig.  89). 


I 

I 

COS  5  Z 

sin  5  z 

y  cos  5  z 

V  sin  5  z 

0° 

9.3 

1.000 

0.000 

9.30 

0.00 

30° 

15.0 

-0.866 

0.500 

-12.99 

7.50 

60° 

17.4 

0.500 

-0.866 

8.70 

-15.07 

90° 

23.0 

0.000 

1.000 

0.00 

23.00 

120° 

37.0 

-0.500 

-0.866 

-18.50 

-32.04 

150° 

31.0 

0.866 

0.500 

26.85 

15.50 

180° 

15.3 

-1.000 

0.000 

'-15.30 

0.00 

210° 

4.0 

0.866 

-0.500 

3.46 

-  2.00 

240° 

-  8.0 

-0.500 

0.866 

4.00 

-  6.93 

270° 

-13.2    - 

0.000 

-1.000 

0.00 

13.20 

300° 

-14.2 

0.500 

0.866 

-  7.10 

-12.30 

330° 

-  6.0 

-0.866 

-0.500 

5.20 

3.00 

2  =             3.62 

-  6.14 

cos  5  xr  =  0.60; 


r  sin  5  *«•  =  —  I-02- 


Hence  the  fifth  harmonic  is  0.60  cos  5  x  —  1  .02  sin  5  x. 

It  is  evident  that  the  labor  involved  in  the  direct  determination  of  the 
coefficients  by  the  above  formulas  is  very  great.  This  labor  may  be 
reduced  to  a  minimum  by  arranging  the  work  in  tabular  form.  These 
forms  follow  the  methods  devised  by  Runge  *  for  periodic  curves  in- 
volving both  even  and  odd  harmonics  (Art.  89),  and  by  S.  P.  Thompson  f 
for  periodic  curves  involving  only  odd  harmonics  (Art.  90). 

89.  Numerical  evaluation  of  the  coefficients.  Even  and  odd  har- 
monics. — 

(I)  Six-ordinate  scheme.  —  Given  the  curve  and  wishing  to  determine 
the  first  three  harmonics,  i.e.,  the  6  coefficients  in  the  equation 

y  =  OQ  +  a\  cos  x  +  02  cos  2  x  +  a3  cos  3  x  +  bi  sin  x  +  bz  sin  2  x, 
we  divide  the  period  from  x  =  o°  to  x  =  360°  t  into  6  equal  parts  and 

*  Zeit.  f.  Math.  u.  Phys.,  xlviii.  443  (1903),  Hi.  117  (1905);  Erlauterung  des  Rech- 
nungsformulars,  u.s.w.,  Braunschweig,  1913. 

t  Proc.  Phys.  Soc.,  xix.  443,  1905;  The  Electrician,  5th  May,  1905. 
t  If  the  period  is  taken  equal  to  2  ir/m  instead  of  2  IT,  the  representing  trigonometric 
series  has  the  form 

y  =  Oo  +  Ci  cos  md  +  a?  cos  2  md  +  •  •  • 
+  bi  sin  md  +  £>2  sin  2  md  +  •  •  •  , 

where  0  represents  abscissas.  By  the  substitution  mO  =  x  or  0  =  x/m,  the  series  be- 
comes 

.  N       V  =  Oo  +  a,\  cos  x  +  02  cos  2  x  +  •  •  •  A 
+  61  sin  x  +  6j  sin  2  x  +  •  •  •  , 

and  this  has  a  period  2  TT.  The  abscissas  from  6  =  o  to  6  =  2  r/m  now  become  the 
abscissas  from  x  =  o  to  x  =  2  IT,  and  we  proceed  to  determine  the  coefficients  in  the 
second  series  as  outlined.  Having  determined  the  coefficients,  we  finally  replace  x  by 

me. 


l8o  EMPIRICAL   FORMULAS  —  PERIODIC   CURVES  CHAP.  VII 

measure  the  ordinates  at  the  beginning  of  each  interval;  let  these  be  rep- 
resented by  the  following  table: 


y 


6oc 


:8oc 


y* 


240 


y* 


Here  n  =  6,  and  using  the  formulas  on  p.  177,  we  have 

=  ^0         +yi  +^2  +ya  +^4 

=  y0          —yi  -\-y-i  —y3  +y* 


3  bz  =  ^osino0  -\-y\  sin  120°  +;y2sin24O°  +y3sin36o°  +^4sin48o°  4-y5sin6oo° 
We  arrange  the  y's  in  two  rows, 

3-0        yi        y*       y* 


Sum          VQ          Vi          1)%          v3 
Diff.  w,         wz 

where  the  z/s  are  the  sums  and  the  w's  are  the  differences  of  the  quantities 
standing  in  the  same  vertical  column ;  thus,  v0  =  y0,  v\  =  y\  +  3*5,  Wi  =  yi 
—  3/5,  etc.  Since  cos  240°  =  cos  120°,  cos  300°  =  cos  60°,  etc.  We  may 
now  write 

6  a0  =  z>o  +  Vi  +  %  +  v 3 

3  ai  =  v0  +  vi  cos  60°  +  v2  cos  120°  +  v3  cos  180° 
3  az  =  v0  +  Vi  cos  120°  +  z/2  cos  240°  +  v3  cos  360° 
3  61  =  Wi  sin  60°  +  w-i  sin  120° 

3  bz  =  Wi  sin  120°  +  ^2  sin  240° 

We  arrange  the  z>'s  and  w's  in  rows, 


Sum  po         Pi  r\ 

Diff.  go          2i  5i 

and  we  now  write 

6ao  =  po  +  pi,  6a3  =  qo  —  q\, 

3  ffi  =  So  +  i  2i,  3  «2  =  po  -  i  pi, 

3*i=  ^rlf  3  *.  =  ^5i. 

Example.     Determine  the  first  three  harmonics  for  the  following  data 
taken  from  Fig.  86&. 


ART.  89  NUMERICAL  EVALUATION  OF  THE  COEFFICIENTS  181 


x           o°             6o° 

120°                 I  80° 

240° 

300° 

,y_       0.47           1.77 
0.47 

2.  2O              —2.  2O 
1.77                2.20 
—  0.49            —1.64 

-1.64 

—  2.20 

-0.49 

v        0.47 

1.28                0.56 

—  2.20 

w 

2.26                3.84 

0.47             1.28 

2.26 

—  2.  2O                0.56 

3.84 

p           -1-73                 1-84 

r            6.10 

q             2.67             0.72 

s         -1.58 

600  =  o.n, 

6a3  =  1.95,            301  =  3.03, 

302  =  -2.65, 

3  bi  =  5-28, 

362  =  -1.37- 

Hence,        OQ  =  0.02,     a\  = 

=  i.oi,      02  =  —0.88, 

as  =  0.33, 

bi  =  1.76,       &2  =  —0.46, 
and  y  =  0.02  +  i.oi  cos  *  —0.88  cos  2  x  +  0.33  cos  3  x 

+  1.76  sin  x  —  0.46  sin  2  jc. 
The  equation  from  which  the  curve  in  Fig.  866  was  plotted  was 


=  cos  x— 0.87  cos  2  £+0.35  cos  3  x + 1. 73  sin  £+0.50  sin  2  #—0.35  sin  3  x. 

We  observe  the  close  agreement  between  the  two  sets  of  coefficients, 
the  small  discrepancies  being  due  to  the  approximate  measurements  of 
the  ordinates  for  our  example. 

(II)   Twelve-ordinate  scheme.  —  Given  the  curve  and  wishing  to  deter- 
mine the  first  six  harmonics,  i.e.,  the  12  coefficients  in  the  equation 
y  =  a0  +  a\  cos  x  +  <h  cos  2  x  +  a3  cos  3  x  +  a\  cos  4  x  -\-  a5  cos  5  x 

+a6  cos  6  x  +  61  sin  x  +  62  sin  2  x  +  63  sin  3  x  +  &4  sin  4  x  +  65  sin  5  #, 
we  divide  the  interval  from  x  =  o  to  x  =  360°  into  12  equal  parts  and 
measure  the  ordinates  at  the  beginning  of  each  interval ;  let  these  be  rep- 
resented by  the  following  table: 


o°    30C 


60°  I  90°  I  120°  I  ISO0  I80°  I  210°  I  240°  270 


y*  \  ys  |   y* 


y* 


yn 


Here  n  =  12,  and  the  formulas  for  the  coefficients  give 

12  Oo  =  y0  +  yi  +  yz  +  •  •  •  +  yu 

12  a6  =  y0  -  yi  +  yz  -   •  •  •   -  yn 

6  a!  =  y0  cos  o°  +  yi  cos  30°  +  3/2  cos  60°    +  •  •  •   +  yn  cos  330° 
6  a2  =  yo  cos  o°  +  ;yi  cos  60°  +  y2  cos  120°  +•••'+  ?u  cos  660° 

6bi  =  y0  sin  o°  +  yi  sin  30°  -f  y2  sin  60°     +  •  •  •  +  yn  sin  330° 
6  &2  =•=  yo  sin  o°  +  3f!  sin  60°  +  yi  sin  120°    +  •  •  •   +  yn  sin  660° 


182 


EMPIRICAL   FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


If  we  arrange  the  y's  in  two  rows 

yo          yi          y-i 


Sum 
Diff. 


VQ 


Wi 


and  remember  that  cos  330°  =  cos  30°,  sin  330°  =  —sin  30°,  etc.,  the 
above  equations  may  be  written 

12  do  =  VQ  ~\~  Vi  4~  Vz  +    '    '    '    +  ^6 

12  d6   =   VQ   —  Vi  +  Vz  —     '    •    •     +  08 

6  ai  =  VQ  +  fi  cos  30°  4-  Vz  cos  60°    4-  •  •  •   +  v6  cos  180° 
6  dz  =  v0  -\-  Vi  cos  60°  +  v2  cos  120°  +  •  •  •   +  v6  cos  360° 


6  &i  =  Wi  sin  30°  +  w2  sin  60°    4-  •  •  •   4~  ^5  sin  150° 

6  62  =  Wi  sin  60°  +  wz  sin  120°  +  •  •  •   +  w5  sin  300° 


If  we  now  arrange  the  t/'s  and  w's  in  two  rows 

fo  Vl  %  Z>3  W 

v&  v$  Vi  w 

Sum          po  pi  pz  PS  T\ 

Diff.          qo  qi  qz  Si 


the  equations  may  be  written 

12  a0  =  qo  4~  <Zi  +22  +  2s 

12  a6  =  po  —  PI  -\-p-i  —  pz 

6  fli  =  50  +  2i  cos  30°  +  qz  cos  60° 

6  a2  =  po  +  />i  cos  60°  4-  pz  cos  120°  +  p3  cos  180° 


6  bi  =  r\  sin  30° 
6  &2  =  ^i  sin  6oc 


+  r-i  sin  60°      +  ra  sin  90° 
+  52  sin  120° 


Finally,  if  we  arrange  the  p's,  g's,  and  r's  as  follows : 


£2 


Pi 

P3 


Sum          /, 
the  equations  become 

12  Oo 


Diff. 


I2fl6    =    IQ    — 


6ai 
6  dz 
6a3 


sin  60°  +  q2  sin  30°.         6  a5  =  g0  —  2i  s'm  60°  +  22  sin  30°. 


-/>2  sn  30 


6  a4  = 
6  63  = 


sin  30 


6  61  =  ri  sin  30°  +  r2  sin  60°  -f-  r3.          6  65  =  r\  sin  30°  —  r2  sin  60°  +  rs. 


6  62  =  (si  +  52)  sin  60°. 


=  (si  —  s2)  sin  60°. 


ART.  89 


NUMERICAL  EVALUATION  OF  THE   COEFFICIENTS 


183 


We  may  now  arrange  the  above  scheme  in  a  computing  form  as  fol- 
lows: 

Ordinates        y0          yi          yz          yz  y*          y&          yt 

yn        yio        y»         y»         yi 


Sum          v{ 

)                Vl 

Vz                 V$                V±                 V  5                 V, 

Diff. 

w\ 

Wz               Wz               W\               W& 

Sum        po 
Diff.        go 

Pi               p2 

2i          2z 

Pi 

p3                                      fl               Ti               7*3 
S\              Sz 

*i          9.0 

Pz              p3                                                                          ?3               ff2 

Sum        lo 

k 

Diff.         /i          tz 

Multipliers  of  the  quan- 
tities in  the  same 
horizontal  rows  before 

Cosine  terms 

Sine  terms 

these  are  entered 

sin  30°  =  0.5 

?2 

-Pi        Pi 

n 

sin  60°  =  0  866 

9i 

r2 

Si        Sl 

sin  90°  =  1.0 

tfo 

Po         -ps 

<2 

la          h 

r» 

ti 

Sum  of  1st  column 

Sum  of  2d  column 

Sum 

6d 

6a2 

6  as 

12  a0 

661 

66, 

66, 

Difference 

6as 

6a4 

12  a. 

6h 

6&4 

Checks :  y0  =  ^o  +  fli  +  CL-L  +  03  +  ^4  +  a&  +  a$. 

yi  -  yu  =  (&i  +  h)  +  \/3  (6,  +  64)  +  2  ft,. 
Result:     y  =  OQ  +  «i  cos  x  +  ag  cos  2  x  +  •  •  •  +  a6  cos  6  x 
+  fti  sin  x  +  bz  sin  2  oc  -J-  •  •  •  +  fts  sin  5  «. 


/r 


FIG.  89. 


i84 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


Example.  In  the  periodic  curve  of  Fig.  89,  the  interval  from  x  =  o° 
to  x  =  360°  is  divided  into  12  equal  parts  and  the  ordinates  y0  to  yn  are 
measured. 


o" 

9-3 


15.0 


60° 

17.4 


90° 

120° 

150° 

1  80° 

210° 

240° 

270° 

300° 

330° 

23.0 

37-o 

31.0 

15-3 

4.0 

-8.0 

-13.2 

-14.2 

-6.0 

We  shall  determine  the  first  six  harmonics  by  the  above  scheme. 
Ordinates       9.3  15.0        17.4        23.0        37.0        31.0       15.3 

—  6.0    —14.2    —13.2      —8.0         4.0 


Sum  (i»)          9.3             9.0          3.2 

9.8       29.0       35.0       15.3 

Diff.  (w) 

21.0          31.6 

36.2       45.0       27.0 

9-3 

9.0 

3.2     9.8 

21.0 

31-6 

36.2 

15-3 

35-0 

29.0 

27.0 

45-0 

Sum  (p)        24.6 

44-o 

32.2     9.8 

W 

48.0 

76.6 

36.2 

\  Diff.  (g)       -6.0 

—  26.0 

-25-8 

(*) 

-6.0 

-13-4 

24.6 

44.0 

48.1 

-6.0 

32.2 

9.8 

36.2 

-25-8 

Sum  (/) 

56.8 

53-8 

Diff.  (/) 

11.9 

19.8 

Multipliers 

Cosine  terms 

Sine  terms 

0.5 
0.866 
t.O 

-12.9 
-22.5 
-6.0 

-16.1        22.0 
24.6      -9.8 

19.8 

56.8    53.8 

24.0 
66.3 
36.2 

-5.2  -11.6 

11.9 

Sum  of  1st  col. 
Sum  of  2d  col. 

-18.9 
-22.5 

8.5 
12.2 

56.8 
53.8 

60.2 
66.3 

-  5.2 
-11.6 

Sum 
Diff.  « 

-41.4  =  60! 
3.6  =  6o6 

20.7  =  6a2 
-3.7=604 

19.8 
=  6a3 

110.  6=12  ac 
3.0=12o6 

126.  5  =  6  6j 
-6.1  =  665 

-16.8=66j 
6.4=664 

11.9 
=  663 

d=— 6.90,  02  =  3.45,03  =  3.30,  a0  =  9.22,  bi  =21.08,    bz=  —2.80,  63  =  1.98, 
05  =  0.60,       #4=—  0.62,  05  =  0.25,  65=  —  i. 02,  64=  1.07. 

Check:    9.3  =  9.22  -  6.90  +  3.45  +  3.30  -  0.62  +  0.60  +  0.25  =  9.30. 
21.0  =  (21.08  —  1.02)  +  1.732  (  —  2.80  +  1.07)  +  2^(1.98)  =21.02. 

Result:  * 

y  =  9.22  —  6.90  cos  x  +  3.45  cos  2  x  +  3.30  cos  3  x  —  0.62  cos  4  x 

+  0.60  cos  5  x  -f  0.25  cos  6  x  +  21.08  sin  x  —  2.80  sin  2  x 

+  1.98  sin  3  x  +  1.07  sin  4  x  —  1.02  sin  5  x, 
or 

y  =  9.22  +  22.18  sin  (x  —  18.12°)  —  4.44  sin  (2  x  —  50.93°) 

+  3.85  sin  (3  x  +  59-04°)  +  1-24  sin  (4  x  -  30.09°) 

—  1. 1 8  sin  (5  x  —  30.47°)  —  0.25  sin  (6  x  —  90°). 

*  The  coefficients  of  the  fifth  harmonic  agree  with  those  found  by  the  direct  process 
in  Art.  88.  The  time  and  labor  spent  in  the  computation  of  all  six  harmonics  by  means 
of  the  above  computing  form  is  much  less  than  that  spent  in  the  determination  of  the 
fifth  harmonic  alone  by  the  direct  process  in  Art.  85. 


ART.  89 


NUMERICAL  EVALUATION  OF  THE   COEFFICIENTS 


The  last  result  was  obtained  by  using  the  relations 

dk  cos  kx  +  bk  sin  kx  =  Ck  sin  (kx  +  <fo) 


where 


and 


tan-'?*. 


(Ill)  Twenty-four-ordinate  scheme.  —  Given  the  curve  and  wishing  to 
find  the  first  12  harmonics,  i.e.,  the  24  coefficients  in  the  equation 
y  =  OQ  +  a\  cos  x  +  az  cos  2  x  +  •  •  •  -4-^12  cos  12  x 
+  61  sin  x  +  bz  sin  2  x  +  •  •  •   +  6U  sin  1 1  x, 

we  divide  the  interval  from  x  =  o°  to  x  =  360°  into  24  equal  parts  and 
measure  the  ordinates  at  the  beginning  of  each  interval ;  let  these  be  repre- 
sented by  the  following  table: 


15' 


330C 


345' 


y        yo     I     yi     \     y*     \     y3     \  .  . 

If  we  use  the  same  method  as  that  employed  in  deriving  the  12-ordi- 
nate  scheme,  we  shall  arrive  at  the  following  24-ordinate  computing  form. 
This  form  is  self-explanatory. 

Ordinates         y0  y\  yz  ...  yn  yu 


Sum                  i 

VQ                Vi                Vz 

...              Vn             fia 

Diff. 

Wi              Wz 

...              wn 

VQ           Vi 

...       v&        v6 

Wi          Wz          ...          W*         W6 

Viz         Vn 

...          V7 

Wn      WIQ      ...       w7 

Sum      po       pi 

...           Pb           p6 

TI        rz        .  .  .       r  6        r6 

Diff.      2o        <Zi 

...       q, 

S{           Sz           ...          55 

Po 

Pi                 p2                 p3 

Si                 Sz                 S3 

P* 

ps            p* 

^6                 ^4 

Sum            IQ 

k       k       k 

ki                 kz                 k3 

Diff.            mo 

mi           mz 

ni           n2 

k           k 

2o  —  54  =  A) 

Tl  +  rs  _  r&  =  Ul 

.'     k          k 

2i  -  2s  -  2s  =  k 

rz-  r6  =  uz 

go          gi 

Multipliers 

Cosine  terms 

Sine  terms 

sin  30°  =0.5' 
sin  60°  =  0.866 
sin  90°  =  1.0 

00            01 

WJ2 

mi 

m0 

-Z.     Zi 
?o-?3 

TOO       TO2 

*'     fc 

*S 

ni       Tiz 

i,      Jfc, 

Sum  of  1st  col. 
Sum  of  2d  col. 

Sum 
Difference 

24  a0 
24ai2 

12  at 
12oip 

12  cu 
12  as 

12  Os 

12  62 
12  610 

1264 
12  68 

12  66 

i86 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


Multipliers 

Cosine  terms 

Sine  terms 

sin  15°  =  0.259 

9s 

9i 

TI 

r6 

sin  30°  =  0.5 

94 

94 

r2 

rj 

sin  45°  =  0.707 

9s 

ti 

—  93 

r3 

Ui 

—  fi 

sin  60°  =  0.866 

92 

-92 

n 

—  r4 

sin  75°  =  0.966 

9i 

95 

r6 

n 

sin  90°  =  1.0 

9o 

U 

9o 

r6 

W2 

r6 

Sum  of  1st  col. 

Sum  of  2d  col. 

Sum 

12fll 

12  a3 

12  a6 

12  61 

1263 

1266 

Difference 

12  a,, 

12  09 

12  a7 

12  61, 

12  b9 

12fe7 

Checks: 


y0  =  a0 


-f  a2  + 


a12. 


Cxi  ~  ^23)  = 


0.259  (& 
+  0.866 


+  0.966  Os 


0.707  (63 

+  67)  -f- 


Result: 


a0  +  &\  cos  x  +  #2  cos  2  x  + 
+  61  sin  ;t  +  b2  sin  2  x  + 


+  #12  cos  12  x 
+  b\\  sin  1  1  x, 


or    y 


ci  sn 


+  c2  sin  (2 


+ 


+Ci2sin  (12  x 


We  shall  now  pass  on  to  the  evaluation  of  the  coefficients  when  only 
the  odd  harmonics  are  present.* 

90.  Numerical  evaluation  of  the  coefficients.  Odd  harmonics  only.  — 
Most  problems  in  alternating  currents  and  voltages  present  waves  where 
the  second  half-period  is  merely  a  repetition  below  the  axis  of  the  first 
half-period;  the  axis  or  zero  line  is  chosen  midway  between  the  highest 
and  lowest  points  of  the  wave  (Fig.  86c).  We  have  shown  in  Art.  86  that, 
in  such  cases,  the  trigonometric  series  contains  only  the  odd  harmonics. 
Furthermore,  since  the  sum  of  the  ordinates  over  the  entire  period  is 

evidently  zero,  then  a0  =  -  V;y  =  o,  and  the  series  does  not  contain  the 
n  •*•» 

constant  term  ao.     Again,  since 

cos  k  (x  -f-  TT)  =  cos  (kx  +  kir)  =  —  cos  kx,  when  k  is  odd, 
sin  k  (x  +  TT)  =  sin  (kx  +  kir)  =  —  sin  kx,  when  k  is  odd, 

and  yx+T  =  —ylt  •'•     yz+*  cos  k  (x  +  TT)  =  yx  cos  kx, 


and  2?  cos  kx  has  the  same  value  over  the  second  half-period  as  over  the 

*  T.  R.  Running,  Empirical  Formulas,  p.  74,  gives  similar  schemes  with  8,  10,  16, 
and  20  ordinates,  for  waves  having  even  and  odd  harmonics.  H.  O.  Taylor,  in  the 
Physical  Review,  N.  S.,  Vol.  VI  (1915),  p.  303,  gives  a  somewhat  different  scheme  with 
24  ordinates  for  waves  having  even  and  odd  harmonics.  A  very  convenient  computing 
form  for  the  above  scheme  with  24  ordinates  has  been  devised  by  E.  T.  Whittaker  for 
use  in  his  mathematical  laboratory  at  the  University  of  Edinburgh;  see  Carse  and 
Shearer,  ibid.,  p.  22. 


ART.  90 


NUMERICAL  EVALUATION  OF  THE   COEFFICIENTS 


I87 


first  half.     Hence  in  finding  the  coefficients  we  need  merely  carry  the 
summation  over  the  first  hali'-period  ;  thus, 


ak  =  -  2?  cos  kx, 


bk  =  -  ^y  sin  kx, 


where  k  is  odd,  x  and  y  are  measured  in  the  first  half-period  only,  and  n 
is  the  number  of  intervals  into  which  the  half-period  is  divided. 

(I)  Odd  harmonics  up  to  the  fifth.—  Given  the  curve  and  wishing  to 
determine  the  coefficients  in  the  equation 

y  =  a\  cos  x-\-  az  cos  3  x-\-a$  cos  5  x-\-bi  sin  x-\-b$  sin  3  x-\-b$  sin  5  x, 
we  choose  the  origin  where  the  wave  crosses  the  axis,  so  that  when  XQ  =  o, 
y0  =  o,  divide  the  half-period  into  6  equal  parts,  and  measure  the  5  ordi- 
nates  y\,  yz,  y$,  y±,  y$.     Thus  we  have 


X 

30° 

60° 

90° 

120° 

150° 

y 

y\ 

yz 

y3 

y4 

y* 

For  the  coefficients  we  have  the  following  equations: 
3  ai  =  yi  cos  30°    +  y2  cos  60°    +  y3  cos  90°    +  y4  cos  120°  +  ys  cos  150°. 
3  a3  =  yi  cos  90°    +  y3  cos  180°  +  y,  cos  270°  +  y*  cos  360°  +  y&  cos  450°. 
3  a5  =  yi  cos  150°  +  y2  cos  300°  +  y3  cos  450°  +  yt  cos  600°  +  y5  cos  750°. 
3  b\  =  yi  sin  30°    +  yt  sin  60°    +  y3  sin  90°    +  y4  sin  120°  +  y&  sin  150°. 
3  63  =  yi  sin  90°    +  y2  sin  180°  +  y3  sin  270°  +  y4  sin  360°  +  y&  sin  450°. 
3  b&  =  yi  sin  150°  +  y2  sin  300°  +  ys  sin  450°  +  y±  sin  600°  +  y5  sin  750°. 
Simplifying  and  replacing  the  trigonometric  functions  by  their  values 
in  terms  of  sin  30°  and  sin  60°,  we  may  write 

3  01  =  CV2  -  yO  sin  30°  +  (yi  -  y5)  sin  60°. 
3  a3  =  —  (yt  —  y4)  sin  90°. 
3  «s  =  (ya  —  yO  sin  30°  —  (3/1  —  y6)  sin  60°. 
3  61  =  (yi  +  ys)  sin  30°  +  (y2  +  y4)  sin  60°  +  y3  sin  90°. 
3  b3  =  (yi  —  y3  +  y&^  sin  90°. 

3  &5  =  (yi  +  ys)  sin  30°  —  (y2  +  y4)  sin  60°  -f  y3  sin  90°. 
We  may  conveniently  arrange  the  work  in  the  following  computing 
form: 


y\      yz 

ys      y4 


Si 


53 


Sum 

Diff.  di       d2 

Checks:  o  =  a\  +  a3  +  a5. 

y3  =  bi  -  b3 


Multipliers 

Cosine  terms 

Sine  terms 

sin  30°  =  0.5 
sin  60°  =  0.866 
sin  90°  =  1.0 

h 

-d2 

Si 
S2 
8s 

Sl        ** 

Sum  of  1st  col. 
Sum  of  2d  col. 

Sum 
Diff. 

3at 
3  as 

3  a, 

36! 
36« 

36, 

Result: 

y  =  a\  cos  x  +  as  cos  3  *  +  a5  cos  5  #  +  &i  sin  x  +  63  sin  3  x  -f-  65  sin  5  x. 


1 88 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


The  following  example  will  illustrate  the  rapidity  with  which  the  co- 
efficients may  be  determined. 

Example.  We  wish  to  analyze  the  symmetric  wave  of  Fig.  900,  i.e., 
to  find  the  coefficients  of  the  1st,  3d,  and  5th  harmonics.  Choose  the 
#-axis  midway  between  the  highest  and  lowest  points  of  the  wave,  and 


FIG.  900. 

the  origin  at  the  point  where  the  wave  crosses  this  axis  in  the  positive 
direction.  Then  divide  the  half-period  into  6  equal  parts  and  measure 
the  ordinates  y\,  .  .  .  ,  yb.  These  are  given  in  the  following  table: 

30°     I      60°     I      90°      I     120°    I     150° 


10.7 


2.8 


20.5 


26.5 


16.6 


We  arrange  the  work  in  the  above  computing  form. 


10.7        2.8    20.5 
16.6      26.5 

Multiplier 

Corine  terms 

Sine  terms 

0.5 
0.866 
1.0 

-11.85 
-5.11 

23.7 

13.65 
25.37 
20.  5 

27.3    20.5 

Sum  of  1st  col. 
Sum  of  2dcol. 

-11.85 
-5.11 

34.15 
25.37 

27.3 
20.5 

Sum  (s)    27.3      29.3   20.5 
Diff.(d)  -5.9  -23.7 

Sum 
Diff. 

-16.96 
-6.74 

23.7 

59.52 

8.78 

6.8 

Divide  by  3 

ai=-5.65 
o6=-2.25 

a,  =  7.9 

6i  =  19.84 
66=  2.93 

fc,  =  2.27 

Check:       ai  +  a3  +  ab  =  -5.65  +  7.90  -  2.25  =  o. 
bi  -  b3  +  bb  =  19.84  -  2.27  +  2.93  =  20.5 


ART.  90  NUMERICAL  EVALUATION  OF  THE   COEFFICIENTS  189 

Result: 

y  =  - 5.65  cos  x  +  7.90  cos  3  x-  2.25  cos  5  x 
+  19.84  sin  x  +  2.27  sin  3  x  +  2.93  sin  5  x. 

(II)  Odd  harmonics  up  to  the  eleventh. — Given  a  symmetric  curve  and 
wishing  to  determine  the  coefficients  in  the  equation 

y  =  ai  cos  x  +  a3  cos  3  x  +  •  •  •   +  an  cos  1 1  x 
+  b\  sin  x  +  b3  sin  3  x  -\-  •  •  •   +  b\\  sin  1 1  x, 

we  choose  the  origin  at  the  point  where  the  wave  crosses  the  axis,  so  that 
y0  =  o,  divide  the  half- period  into  12  equal  parts,  and  measure  the  II 
ordinates  yi,  yz,  .  .  .  ,  yn-  Thus  we  have 


X 

15° 

30° 

45° 

165° 

y 

yi 

y* 

ys 

yn 

For  the  coefficients  we  have  the  following  equations : 

6  ai  =  yi  cos  15°  +  yz  cos  30°  +  •  •  •   +  yu  cos  165°. 
6  a3  =  yi  cos  45°  +  y-2  cos  90°  +  •  •  •  +  yu  cos  495°. 

6  bi  =  yi  sin  15°  +  y2  sin  30°  +  •  •  •   +  yn  sin  165°. 
6  63  =  y\  sin  45°  +  yz  sin  90°  +  •  •  •   +  yu  sin  495°. 

If  we  arrange  the  ordinates  in  two  rows, 

y\       yz       ys       yi       y*>       y& 


Sum  Si          $z          s3          54          Ss          s$ 

Difi.  di         dz         d3         di         d$         d$ 

replace  the  trigonometric  functions  by  their  values  in  terms  of  the  sines 
of  15°,  30°,  45°,  60°,  75°,  90°,  and  collect  terms,  we  may  write 

6ai  =     4sini5°+d4sin3O0-f-<f3sin450-H/2sin6o0+disin75°. 
6  an=  —d&  sin  15°+^  sin  30°  —  J3  sin  45° +^2  sin  60°  —  disin75°. 
6  a5  =     di  sin  15°+^  sin  30°  —  d3  sin  45°  —  dz  sin  6o°+J5  sin  75°. 
6  a7  =  —di  sin  15°+^  sin  30°+^  sin  45° -dz  sin  60° -d6  sin  75°. 
6  bi  =     Si  sin  15°+^  sin  30° +^3  sin  45°+^  sin  6o°+s&  sin  75°+s6  sin  90°. 
6  6n=     Si  sin  15°  —  Sz  sin3O°+s3  sin  45°  —  st  sin6o°+56  sin  75°  —  -S6 sin 90°. 
666  =     s*,  sini5°+52  sin  30°  —  s3  sin  45°  —  s*  sin6o°+$i  sin  75°+s6  sin  90°. 
6  bi  =     s$  sin  15°  — s2  sin  30°— s3  sin  45° +$4  sin6o°+Si  sin  75°— 56  sin  90°. 
6  a3  =  (di  —  d3  —  <£5)  sin  45°  —d*  sin  90°. 

6  a%  =  —  (di  —  d3  —  0*5)  sin  45°  —d\  sin  90°. 

+  £3  —  *s)  sin  45°  +  (^2  —  s6)  sin  90°. 

—  55)  sin  45°  —  ($2  —  s6)  sin  90°. 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


We  may  conveniently  arrange  the  work  in  the  following  computing 
form: 

y\     y%     y$    y^    y$    yo  ^1+^3  —  s&  =  f\ 

yu    yio    y$    y&    y^  ^2  —  ^e         ==  ft 

Diff.      di      d*     d3      di     d& 


Multipliers 

Cosine  terms 

Sine  terms 

sin  15°  =  0.259 

d, 

di 

Si 

>i 

sin  30°  =  0.5 

d* 

d4 

St 

Sl 

sin  45°  =  0.707 

d3 

e\ 

-d* 

83 

Tl 

—  st 

sin  60°  =  0.866 

dt 

-d2 

«4 

—  Si 

sin  75°  =  0.966 

di 

d6 

Si 

Si 

sin  90°  =  1.0 

-d< 

S6 

r* 

it 

Sum  of  1st  col. 

Sum  of  2d  col. 

Sum 

6d 

6  as 

6a5 

661 

66, 

666 

Diff. 

6  aa 

6  a9 

6  a7 

6611 

669 

6b7 

Checks: 


+  an  =  O, 


Result :     y  =  ai  cos  x  +  a3  cos  3  x  +  •  •  •   +  an  cos  1 1  x 
+  b\  sin  x  +  b3  sin  3  x  +  •  •  •   +  b\\  sin  1 1  x. 


50- 


I 


15°  30°  45°  60°  75°  90°  105°  120°  136°  150°  165° 


FIG.  906. 


Example.  Fig.  906  represents  a  half-period  of  an  e.m.f.  wave  whose 
frequency  is  60  cycles.  We  wish  to  find  the  odd  harmonics  up  to  the  nth 
order.  Choose  the  x-axis  midway  between  the  highest  and  lowest  points 
of  the  complete  wave  and  the  origin  at  the  point  where  the  wave  crosses 
the  #-axis  in  the  positive  direction.  Divide  the  half-period  into  12  equal 


ART.  90 


NUMERICAL   EVALUATION  OF  THE   COEFFICIENTS 


parts  and  measure  the  ordinates  yi,  y%  .  .  .  ,  yn.     These  are  given  in 
the  following  .table: 
x    I  15°  I  30°     45°     60°  I  75°     90°    105°    120°     135°  I  150°  I  165° 


30 


y    I    4    I   21        19  t    27    |   29       33       46       38        50 
We  arrange  the  work  in  the  above  computing  form. 

4    21     19    27     29     33  37  +  69  -  75  =  31 

33     30    50    38    46  51-33 

Sums  (s) 


33 


37     51 
Diff.  (d)   —29—  9 


69     65     75 
31  —  11  —  17 


33 


=  18  = 
-29  +  31  +  17  =  19  = 


Multipliers 

Cosine  terms 

Sine  terms 

0.259 

0.5 

0.707 
0.866 
0.966 
1.0 

-4.4 

-5.5 
-21.9 

-7.8 
-28.0 

13.4 
11.0 

-7.5 
-5.5 
21.9 

7.8 
-16.4 

9.6 
25.5 
48.8 
56.3 
72.5 
33.0 

21.9 
18.0 

19.4 
25.5 
-48.8 
-56.3 
35.7 
33.0 

Sum  1st  col. 
Sum  2d  col. 

-13.3 
-54.3 

11.0 
13.4 

2.3 
-2.0 

130.9 
114.8 

21.9 
18.0 

6.3 
2.2 

Sum 
Diff. 

-67.6 
41.0 

24.4 
-2.4 

0.3 
4.3 

245.7 
16.1 

39.9 
3.9 

•    8.5 
4.1 

Divide  by  6 

d  =-11.27 
au=      6.83 

a3=      4.07 
a9  =  -0.40 

05  =  0.05 
07  =  0.72 

61  =40.95 
6,1=  2.68 

63  =  6.  65 
69  =  0.  65 

65  =  1.42 
67=0.68 

•  +an=  -11.27+4.07+0.05+0.72-0.40  +  6.83  =  0 

•  -bn  =  40.95  -6.65  +  1.42  -0.68  +0.65  -2.68  =  33.01= 


Check: 
ai+a3+  • 
bi-b3+  - 
Result: 

y  =  —  1  1  .27  cos  x  +  4.07  cos  3  x  +  0.05  cos  5  x  +  0.72  cos  7  x  —  0.40  cos  9  x 
+  6.83  cos  1  1  x  +  40.95  sin  x  +  6.65  sin  3  #  +  1  .42  sin  5  x 
+  0.68  sin  7  x  +  0.65  sin  9  x  +  2.68  sin  1  1  x. 

(Ill)  Odd  harmonics  up  to  the  seventeenth.  —  Given  a  symmetric  curve 
and  wishing  to  determine  the  coefficients  in  the  equation 

y  =  a\  cos  x  +  0,3  cos  3  x  +  •  •  •  +  an  cos  1  7  x 
+  bi  sin  x  +  bs  sin  3  x  +  •  •  •  +  bn  sin  17  x, 

we  choose  the  origin  at  the  point  where  the  wave  crosses  the  axis,  so  that 
yo  =  o,  divide  the  half  -period  into  18  equal  parts,  and  measure  the  17 
ordinates  yi,  yt,  .  .  .  ,  yn.  Thus  we  have 


X 

10° 

20° 

30° 

170° 

y 

y\ 

y* 

y3 

.  .   . 

yn 

If  we  use  the  same  method  as  that  employed  in  deriving  the  n-ordi- 
nate  scheme,  we  shall  arrive  at  the  following  17-ordinate  computing  form. 
This  form  is  self-explanatory.  '  '  »  , 


192  EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 

yn      y\6      yi&      yu.      y\s      yu      yn 


CHAP.  VII 


Sum        $i 

Sz 

S3 

S^              Si 

Sfi               S7 

s»          s* 

Diff.        dl 

d* 

dz 

d,      d 

5          d,          d7 

d, 

Si 

s* 

S3 

r\ 

dz     d, 

—  de            —  £1 

-si 

S4 

~S9 

-rs 

-d,    -d, 

C3 

Sum 


Multipliers 

Cosine  terms 

Sine  terms 

sin  10°  =  0.1737 
sin  20°  -0.3420 
sin  30°  0.5000 
sin  40°  0.6428 
sin  50°  0.7660 
sin  60°  0.8660 
sin  70°  0.9397 
sin  80°  0.9848 
sin  90°  =  1.0000 

d, 
d< 
d6 
ds 
d* 
da 
d2 
d, 

Ci 

e-i 
e3 

-dt 
-d* 

d6 
di 
di 
-d> 
-dt 
d7 

d< 

di 
d, 
-d7 
-dz 
-d3 
-di 
d. 

e\ 

S2 
83 
S4 
S5 

Se 
s^ 

Si 

sg 

Ti 

r-t 

r.-i 

—  s; 

—  84 
S3 
Si 

—st 

S2 

—  Si 

—s, 

—  83 
S2 

s? 
Se 

—  «4 

r* 

Sum  of  1st  col. 
Sum  of  2d  col. 

Sum 
Diff. 

90l 

9  an 

9o3 

9a15 

9a6- 
9a13 

9a7 
9  an 

9fl9 

9  61 

9&17 

90s 
9616 

9  65 
96,3 

9  67 
96n 

96, 

Check: 


a\  +  a3  +  a5  +  • 


=  o, 


Result :       y  =  a\  cos  x  +  a3  cos  3  x  +  •  •  •   +  an  cos  1 7  x 

+  bi  sin  x  +  63  sin  3  x  +  •  •  •   +  bn  sin  1 1  x. 

Similar  computing  forms  may  be  constructed  for  symmetrical  waves 
containing  odd  harmonics  up  to  the  seventh,  ninth,  etc.,  orders. 

91.  Numerical  evaluation  of  the  coefficients.  Averaging  selected 
ordinates.*  —  We  are  to  determine  the  coefficients  in  the  trigonometric 
series 

y  =  a0  +  ai  cos  x  +  az  cos  2  x  +  •  •  •   -f-  a*  cos  kx  -\-  •  •  • 
+  bi  sin  x  +  b%  sin  2  x  +  •  •  •   +  bk  sin  kx  +  •  •  •  • 

Let  an  and  bn  represent  the  coefficients  of  any  harmonic.     We  divide 
the  period  2  IT  into  n  equal  intervals  of  width  2  ir/n  and  measure  the  ordi- 
nates at  the  beginning  of  these  intervals.     We  have  the  table 
x     I       x0 


*  These  methods  have  been  developed  by  J.  Fischer-Hinnen,  Elekrotechm'sche 
Zeitschrift,  May  9,  1901,  and  S.  P.  Thompson,  Proc.  of  the  Phys.  Soc  .  of  London, 
Vol.  XXIII,  1911,  p.  334.  See,  also,  a  description  of  the  Fischer-Hinnen  method  by 
P.  M.  Lincoln,  The  Electric  Journal,  Vol.  5,  1908,  p.  386. 


ART.  91  NUMERICAL  EVALUATKJN  OF  THE   COEFFICIENTS  193 

Substituting  these  pairs  of  values  in  our  series,  we  have  n  equations 
of  the  form 

yr  =  00  +  0,i  COS  XT  +  02  COS  2  Xr  +    •    •    •    +  dk  COS  kxr  +    -    •    • 

+  bi  sin  xr  +  bz  sin  2  xr  +  •  •  •  +  br  sin  kxr  -K  •  •  •  , 

where  r  takes  in  succession  the  values  o,  1,2,3,  ...,n—  i;  adding 
these  n  equations,  we  get 


where  the  summation  is  carried  from  r  =  otor  =  n—  I. 

If   we    let   /3  =  jfe   —   in    the   expressions   for  V  cos   (a  +  r|8)   and 
w 

2  sin(a  +  r|8)  derived  in  the  note  on  p.  175,  these  become 

•v^       /       ,    2ir\        sin  kir  /     .  k(n  —  I)TT\  .         .    . 

>,cos  a+£r  —   =  -  —  ,,     .   .cos  <H  --  -  1  =  O,  since  sin  kir  =  o, 

^      \  n  )     sin  (Jbr/tt)        \  n        } 

•^\  .     I     .  ,    2ir\          sin  kir        .     I      .  k(n—  I)TT\  .  .     , 

Vsm  I  a-\-kr  —  )=  -s  —  j-,  —  r^sin  \a-\-  —  -)  =  o,  since  sin  kir  =  o, 

^       \  n  I     sin  (Jhr/tl)        \  n        / 

except  when  £  is  a  multiple  of  n,  for  then  both  sin  kir  and  sin  (kir/n)  are 
equal  to  zero  and  the  fractional  expression  becomes  indeterminate  But 
when  k  is  a  multiple  of  «, 

2)cosf  a  +  &r  -  —  j  =  ^cos  (a  +  multiple  of  2x)  =  ^cosa  =  wcos  a. 

^sin  (a-\-kr  —  j  =  ^sin  (a  +  multiple  of  2  T)  =  ^sin  a  =  n  sin  a. 
Hence  we  may  state 

^cos  (  a  +  kr  —  j  =  o,  except  when  k  =  n,  2  n,  3  n,  .  .  . 
=  n  cos  a,  when  k  =  n,  2  n,  3  «,  .  .  .  . 
^sinf  a  +  kr  —  J  =  o,  except  when  k  =  n,  2  n,  3  n,  .  .  . 
=  n  sin  a,  when  k  =  n,  2  n,  3  n,  .  .  .  . 

(l)  If  we  start  our  intervals  at  XQ  =  o,  then  xr  =  r  —  ,  and 

n 

2)cos  kxr  =  ]£}cosf  o  +  kr  —  j  =  o,  except  when  k  =  n,  2  n,  3  n,  .  .  . 

=  n  cos  o  =  n,  when  k  =  n,  2  n,  3  n,  .  ... 
]^sin  kxr  =  2)sin  f  o  +  kr  —  J  =  o,  for  all  values  of  k. 

•'•          ^r  =  wao  +  Man  +  vwa2n  +  •  •  •   —  n  (a0  + 


194 


EMPIRICAL   FORMULAS  —  PERIODIC   CURVES  CHAP.  VII 


(2)  If  we  start  our  intervals  at  x0'  =-,  then  */  =  -  +  r  —  and 

n  n          n 

/  =  y)cos(£-+£r  —  )  =  o  except  when  k  =  n,  2  n,  3  n,  .  .  . 

^^      \  n          HI 


kir\  =  «  when  k  =  2  n,  4«.  6«,  . 
wcos  —  \ 

n  [=  —  n  when  k  =  n,  3  n,  5«,  .  .  . 


sin  kxr'  =  ^sm(k  -  +  kr  —  -)=  o 


for  all  values  of  k. 


=  n  (a0  -  an  +  a2n  -  a3 
Subtracting  the  second  set  of  ordinates,  yf,  from  the  first  set,  y,  we 
have 


or     an+asn+a5n+ 


=  2  n  (an  +  a3r 
j 

2  n  n~          n~    ' 

The  first  set  of  w  ordinates  start  at  x  =  o  and  are  at  intervals  of  2  TT/W, 
and  the  second  set  of  n  ordinates,  start  at  x  =  ir/n  and  are  at  intervals  of 
2  TT/W  ;  thus,  the  period  from  x  =  oto:x;  =  27ris  divided  into  2  n  equal 
parts  each  of  width  ir/n  (Fig.  910).  Hence, 

If,  starting  at  x  =  o,  we  measure  2  n  ordinates  at  intervals  of  ir/n,  the 
average  of  these  ordinates  taken  alternately  plus  and  minus  is  equal  to  the 
sum  of  the  amplitudes  of  the  nth,  3  nth,  5  nth,  .  .  .  cosine  components. 


FIG.  910. 


Thus,  to  determine  the  sum  of  the  amplitudes  of  the  5th,  I5th,  25th, 
.  .  .  cosine  components,  merely  average  the  10  ordinates,  taken  alter- 
nately plus  and  minus,  at  intervals  of  180°  H-  5  =  36°,  or  at  o°,  36°,  72°, 
.  .  .  ,  324°  (Fig.  9ic);  therefore 

«6  4-  0i5  +  OK  +  •  •  •  =  1*5  Cyo  —  ys6  +  yii  —  ym  4-  ^144  —  y 

-  ^252  +  ^288  -  ^324). 


ART.  91  NUMERICAL  EVALATUION  OF  THE   COEFFICIENTS  195 

If  the  1  5th,  25th,  .  .  .  harmonics  are  not  present,  then 

05  =  TV  CVO  —  y&,  +  yn  —  yiOA  +    ;Vl44  —  ym  +  ^216     ~  ^252  +^288—    ^324). 

(3)  Similarly,  if  we  start  our  intervals  at  x0  =  —  ,  then  xr  =  —  +  r  —  , 

2  n  2  n         n 

and 


kxr  =  Scos  (k  —  +  kr  —  )  =  o  for  all  values  of  k, 
^*       \   2  n  n  I 

=  ^sin  f  k  —  +  kr  —  J  =  o  except  when  k  =  n,  2  n,  3  n,  .  .  . 


f  =      n  when  k  =     n,  5  «,    9  n,  .  .  . 
=  w  sin  —  J  =      o  when  k  =  2  n,  4  n,    6  n,  .  .  . 
—  n  when  £  =  3  n,  7  w,  1  1  n,  .  .  . 


(4)  Again,    if   we    start    our    intervals    at   #</  =  --  1  —  ,  then 


+  r 
2  n          n 


^  +  Jfer—  )= 
2  n  n  / 


for  all  values  of  k, 


--  -\-kr  —  -)=  o  except  when  k  =  n,  2  n,  3  n,  .  .  . 
2  n          n  / 

—n  when  k  =  n,  $n,  9  n,  .  .  . 
owhen  k  =  2n,  $n,  6n,  .  .  . 
n  when  k  =  3  n,  7  n,  1  1  n,  .  .  . 

=n(a0-bn+b3n-b5n+bln-  •  -  •  ). 

Subtracting  the  second  set  of  ordinates,  y't  from  the  first  set,  y,  we 
have 


&7n+    '    '    '     =  —  (jO~  >'  +3?1  —  yi+    '    '    '    +^-1  -^'n-l)- 

The  first  set  of  n  ordinates  start  at  x  =  ir/2  n  and  are  at  intervals  of 

2  ir/n,  and  the  second  set  of  n  ordinates  start  at  x  —  --  (-  -  and  are  at 

2  n      n 

intervals  of  2  tr/n  ;  thus  the  period  from  x  =  ir/2  n  to  x  =  2  IT  +  ir/2  n 

is  divided  into  2  n  equal  parts  each  of  width  -•     Hence, 

n 

If,  starting  at  x  —  ir/2  n,  we  measure  2  n  ordinates  at  intervals  of  ir/n, 
the  average  of  these  ordinates  taken  alternately  plus  and  minus  is  equal  to  the 


196 


EMPIRICAL   FORMULAS  —  PERIODIC    CURVES 


CHAP.  VII 


sum  of  the  amplitudes,  taken  alternately  plus  and  minus,  of  the  nth,  3  nth, 
5  nth,  .  .  .  sine  components. 

Thus  to  determine  the  sum  of  the  amplitudes,  taken  alternately 
plus  and  minus,  of  the  5th,  I5th,  25th,  .  .  .  sine  components,  merely 
average  the  10  ordinates  taken  alternately  plus  and  minus,  at  intervals 
of  180°  -r-  5  =  36°,  starting  at  x  =  180°  -5-  10  =  18°,  i.e.,  at  x  =  18°, 
54°,  90°,  .  .  .,  342°  (Fig.  9ic);  therefore 

b&  -  6i6  +  625  -   •  •  •    =  TV  CXis  -  3*54  +  3*90  -  3*126  +  3*162  -  ym  +  3*234 

~   3*270  +  3*306  -  ^42). 

If  the  I5th,  25th,  .  .  .  harmonics  are  not  present,  then 
&5  =  lv  (3*18  -  3*&4  +  3*90  -  ym  +  ym  -  ym  +  3*234   -  3*270  +  3*306  -  3*342). 

We  may  also  note  that  the  set  of  2  n  ordinates  measured  for  deter- 
mining the  6's  lie  midway  between  the  set  of  2  n  ordinates  measured  for 
determining  the  a's,  so  that  to  determine  any  desired  harmonic  we 
actually  measure  4  n  ordinates,  starting  at  x  =  o  and  at  intervals  of 
7T/2  «.  We  use  the  1st,  3d,  5th,  ...  of  these  ordinates  for  determining 
a,  and  the  2d,  4th,  6th,  ...  of  these  ordinates  for  determining  b. 

If  the  higher  harmonics  are  present,  these  must  be  evaluated  first. 
The  absolute  term  a0  is  obtained  from  the  relation 

yo  =  a0  +  a\  +  a*  +  as  +  •  •  •  . 

We  shall  now  illustrate  the  methods  developed  by  an  example. 
Example.     Given  the  periodic  wave  of  Fig.  89  and  assuming  that  no 
higher  harmonics  than  the  6th  are  present,  we  are  to  determine  the  co- 
efficients in  the  equation 

y  =  a0  +  fli  cos  x  +  o2  cos  2  x  +  •  •  •   +  a6  cos  6  x 
+  b\  sin  x  +  bz  sin  2  x  +  •  •  •   -f  b6  sin  6  x. 

To  determine  a6  and  b6  measure  12  ordinates  at  intervals  of  30°  be- 
ginning at  x  =  o°  and  x  =  15°  respectively  (Fig.  916);  then 


FIG. 


ART.  91 


NUMERICAL  EVALUATION  OF  THE   COEFFICIENTS 


197 


=  T*  (jo  -  yso  +  iyeo  -  ysx)  +  '  •  •   +  ym  -  ym) 

=  A  (9-3  -  15-0  + 174  -  23.0  +  37.0  -  31.0  + 15.3  -  4.0  -  8.0  + 13.2 

-  14.2  +  6.0)  =  0.25. 

=  iV  (yi5  —  ^45  +  yib  —  ym  +  •  •  •  +  yzu,  —  y^b) 
=  rV  (13-0  -  16.0  +  19.5  -  31.0  +  35.3  -  23.8  +  10.5  +  5.7  -  10.0 
+  14.5  -  11.0-0.5)  =  0.52. 


30- 


70, 


234°         270° 


-10- 


FIG.  9  ir. 

To  determine  a5  and  65  measure  10  ordinates  at  intervals  of  36°, 
beginning  at  x  =  o°  and  x  =  18°  respectively  (Fig.  91^)  then 

0-5  =  iV  (yo  -  ^36  +  yn  -  yios  +  •  •  •   +  ym  -  ym) 

=  iV  (9-3-I5-3-I-I8.8-32.8+33.0-I5.3-I.O+9.5-I5-0+84) 

=  —0.04. 
b&  =  TV  (yis  —  yiA  +  yw  —  ym  +  •  •  •  +  ym  —  ywt) 

=  TV  (13-8-16.8+23.0-36.8+25.5-9.0-7.7+13.4-13.2+1.5) 

=  -0.63. 


FIG.  gid. 


198  EMPIRICAL   FORMULAS — PERIODIC   CURVES  CHAP.  VII 

To  determine  a4  and  bi  measure  8  ordinates  at  intervals  of  45°,  be- 
ginning at  x  =  o°  and  x  =  22\°  respectively  (Fig.  91^);  then 

04  =  I  (y*  -  y^  +  ^90  -  ym  +  •  •  •  +  ym  -  y>K) 

=  I  (9-3  -  16.0  +  23.0  -  35.3  +  15.3  +  5.7  -  13.2  4-  ii.o)  =  -0.03. 

&4    =    |  (^22.5  —  ^67.5  +  ^112. 5  —     •     •     •     +  ^292.5  ~  ^SST.s) 

=  I  (H-S  -  18.0  +  35.0  -  27.7  +  7.7  +  8.8  -  14.7  +  3.0)  =  i .08. 

To  determine  a3  and  b$  measure  6  ordinates  at  intervals  of  60°,  be- 
ginning at  x  =  o  and  x  =  30°  respectively  (Fig.  916);  then 

as  =  i  (yo  -  yw  +  yuo  -  ym  +  ym  -  ym) 

=  I  (9-3  -  174  +  37-0  -  15-3  -  8.0  +  14.2)  =  3.30. 

bz    =    \  (^30  —  ^90  +  ym  —  yzio  +  ^270  ~  ^SSo) 

=  i  (i5-0  -  23.0  +  31.0  -  4.0  -  13.2  +  6.0)  =  1.97. 

To   determine  a^  and  bz  measure  4  ordinates  at  intervals  of  90°, 
beginning  at  x  =  o°  and  x  =  45°  respectively  (Fig.  916);  then 
02  +  a6  =  I  (yo  -  :V9o  +  ym  -yvo)  =  I  (9-3  -  23.0  +  15.3  +  13-2)  =  3-7O, 

.-.     02  =  345- 

b*  -  b6  =  \  (y&  -  ym  +  3^25  -  y3is)  =  I  (16.0  -  35.3  -  5.7  +  11.0)  =  -3.50, 

.-.     62  =  -2.98. 

To  determine  a\  and  bi  measure  2  ordinates  at  intervals  of  180°, 
beginning  at  x  =  o°  and  x  =  90°  respectively  (Fig.  916);  then 

an  +  a3  +  a&  •=  \  (yQ  -  ym)  =  \  (9.3  -  15.3)  =  -3.00,     /.    ai  =  -6.26. 
bi  -  b3  +  b5  =   £  (y90  -  ym)  =\  (23.0  +  13.2)  =  18.10,      /.     61  =  20.60. 

To  determine  a0  we  have 

a0  +  ai  +  a2  +  as  +  a4  +  a&  +  fle  =  yo  =  9-3.       •*•     <*<>  =  8.63. 
Result: 

y  =  8.63  —  6.26  cos  x  +  3.45  cos  2  x  -\-  3.30  cos  3  x  —  0.03  cos  4  x 
—  0.04  cos  5  x  +  0.25  cos  6  x  +  20.60  sin  re  —  2.98  sin  2  x 
+  i  .97  sin  3  x  +  i  .08  sin  4  x  —  0.63  sin  5  x  +  0.52  sin  6  re. 

This  result  agrees  quite  closely  with  that  of  Art.  89,  p.  184;  the  differ- 
ences in  the  values  of  the  coefficients  are  due  to  the  fact  that  by  the 
method  of  Art.  89  only  the  ordinates  at  o°,  30°,  60°,  .  .  .• ,  330°  are  used, 
whereas  by  the  method  of  this  Art.  a  large  number  of  intermediate  ordi- 
nates are  used.  If  the  curve  is  drawn  by  some  mechanical  instrument, 
the  present  method  will  evidently  give  better  approximations  to  the 
values  of  the  coefficients;  but  the  labor  involved  in  using  the  computing 
form  on  p.  183  is  much  less  than  that  used  in  measuring  the  selected 
ordinates  above. 

92.  Numerical  evaluation  of  the  coefficients.  Averaging  selected 
ordinates.  Odd  harmonics  only.  —  If  the  axis  is  chosen  midway  between 
the  highest  and  lowest  points  of  the  wave  and  the  second  half-period  is 


ART.  92 


NUMERICAL  EVALUATION  OF   THE   COEFFICIENTS 


merely  a  repetition  below  the  axis  of  the  first  half-period,  then  only  the 
odd  harmonics  are  present.  If  the  ordinates  at  x  =  xr  and  x  =  xr  +  TT 
are  designated  by  yr  and  yr+v  respectively,  then  yr+K  =  —  yr.  In  the 
method  of  averaging  selected  ordinates,  the  2  n  ordinates  are  spaced  at 
intervals  of  ir/n  and  are  taken  alternately  plus  and  minus;  then  yr+T  is 
at  a  distance  TT  =  n  (ir/ri),  or  n  intervals,  from  yr,  and  since  n  is  odd,  yr+r 
will  occur  in  the  summation  with  sign  opposite  to  that  with  which  yr 
occurs,  so  that,  e.g. 


+ 


-  v  + 


±3V 


I 

2W 


-  y0+w  +  ;y1+;'  - 
'+  .  .  .   ±2 yr  • 


=  -(3^-yi'+  •  •  •  ±3v  •  •  •  ). 

n 

Hence  we  need  merely  divide  the  half-period  into  n  equal  intervals  and 
average  n  ordinates.  We  may  therefore  restate  our  rules  for  determining 
the  coefficients  if  the  wave  contains  odd  harmonics  only. 

If,  starting  at  x  =  o,  we  measure  n  ordinates  at  intervals  of  ir/n,  the 
average  of  these  ordinates  taken  alternately  plus  and  minus  is  equal  to  the 
sum  of  the  amplitudes  of  the  nth,  3  nth,  5  nth,  .  .  .  cosine  components. 

If,  starting  at  x  =  ir/2  n,  we  measure  n  ordinates  at  intervals  of  TT/W,  the 
average  of  these  ordinates  taken  alternately  plus  and  minus  is  equal  to  the 
sum  of  the  amplitudes,  taken  alternately  plus  and  minus,  of  the  nth,  3  nth, 
5  nth,  .  .  .  sine  components. 

Furthermore,  a0  =  o  since  the  sum  of  the  ordinates  over  the  entire  period 
is  zero. 


FIG.  92. 

Example.  Assuming  that  the  symmetric  wave  of  Fig.  92  contains  no 
higher  harmonics  than  the  5th,  we  are  to  determine  the  1st,  3d,  and  5th 
harmonics.  Applying  the  above  rules  we  have 


EMPIRICAL   FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


05  =  i  (yo  -  y»6  +  yn  -  y 

=  J  (o  —  8.6  +  6.3  —  27.7  +  19.0)  =  —2.20. 

b6  =  I  (yis-yM+ygo-ym+yiwH  i  (11.3-2.7+20.5-25.5+10.7)  =  2.86. 
&s  =  %  (yo  -  yeo  +  yw)  =  l(o  -  2.8  +  26.5)  =  7.90. 
b3  =  %  (yso  -  yw  +  yiso)  =  ?  (10.7  -  20.5  +  16.6)  =  2.27. 
a\  +  a3  +  a5  =  i  (yo)  =  o,  .*.     ai  =  —5.70. 

bi  -  b3  +  65  =  1  (y»)  =  20.5,          .'.     &!  =  +19.91. 

Result: 

y  =  —  5.70  cos  x  +  7.90  cos  3  x  —  2.20  cos  5  x 
+  19.91  sin  x  +  2.27  sin  3  x  +  2.86  sin  5  x. 

We  may  compare  this  result  with  that  obtained  for  the  same  curve 
by  the  use  of  the  computing  form  on  p.  187. 

If  only  the  1st  and  3d  harmonics  had  been  present  in  the  above  wave, 
we  should  have 

as  =  $  (yo  —  y<»  +  yiaO  ;  b3  =  I  (y3o  —  yw  +  yiw)  ; 

01  +  «3  =  yo  =  o;  61  -  63  =  ysx). 

If  all  the  odd  harmonics  up  to  the  ninth  had  been  present  in  the  above 
wave,  we  should  have 

09  =  i  (yo  —  y2o  +  yio  —  yeo  +  yso  —  yioo  +  y\™  —  ym  +  yi«0  ; 
69  =  i  (yio  -  yso  +  yso  -  y?o  +  y9o  -  yno  +  yiao  -  yi5o  +  ym)  ; 

«7    =    j  (yo  —  y25.71  +  ysi.43   ~  >'77.14 


=  ^  (yi2.se  —  yss.57  + 
=  i  (yo  -  yse  +  y?2  - 
+  ^9  =  1  (yo  —  y 


yns.71  —  yi4i.4s  +  yieT.iO  ; 
65  =  J  (yw  -  ys4  +  y^  -  yiae  +  y 
9  =  |  (j3o  —  yoo  +  yiw)  ; 


+  yi44) 


yo  =  o;      i  — 

Similar  schedules  may  be  formed  for  determining  the  odd  harmonics 
up  to  any  order. 

93.  Graphical  evaluation  of  the  coefficients.  —  Various  graphical 
methods  have  been  devised  for  finding  the  values  of  the  coefficients  in 
the  Fourier's  series,  but  these  are  less  accurate  and  much  more  laborious 
than  the  arithmetic  ones.  The  graphical  methods,  while  interesting,  are 
of  little  practical  value  in  rapidly  analyzing  a  periodic  curve,  so  that  we 
shall  describe  here  only  one  of  these  methods  —  the  Ashworth-Harrison 
method.* 

If,  for  example,  we  divide  the  complete  period  into  12  equal  intervals 
and  measure  the  12  ordinates,  we  shall  have  the  table 


0 

30 

60° 

90 

120° 

150 

1  80° 

210° 

240° 

270 

300 

330 

yo 

yi 

y2 

ys 

y4 

ys 

ye 

y7 

ys 

yg 

yio 

yn 

*  Electrician,  Ixvii,  p.  288,  1911;  Engineering,  Ixxxi,  p.  201,  1906.  Other  methods 
are  briefly  mentioned  and  further  references  are  given  in  Modern  Instruments  and 
Methods  of  Calculation,  a  handbook  of  the  Napier  Tercentenary  Celebration. 


ART.  93  GRAPHICAL  EVALUATION  OF   THE   COEFFICIENTS 

We  have  already  shown  (p.  181)  that 
6  ai  =  ^.yr  cos  xr  =  y0  cos  o°  +  yi  cos  30°  +  •  •  •  +  yn  cos  330°, 


201 


6bi 


r  sin  xr  =  y0  sin  o°  +  y\  sin  30° 


+  yu  sin  330°. 


It  is  evident  that  if  we  consider  the  y's  as  a  set  of  co-planar  forces 
radiating  from  a  common  center  at  angles  o°,  30°,  60°,  .  .  .  ,  the  sum  of 
the  horizontal  components  is  equal 
to  6  ai  and  the  sum  of  the  vertical 
components  is  6  b\.  To  facilitate  the 
finding  of  these  sums  we  may  draw 
the  polygon  of  forces,  starting  at  a  ' 
point  0  and  laying  off  in  succession 
the  ordinates,  each  making  an  angle 
of  30°  with  the  preceding,  as  in  Fig. 
930  (proper  regard  must  be  had  for 
the  signs  of  the  ordinates).  The 
polygon  of  forces  may  be  constructed 
rapidly  by  means  of  a  protractor 
carrying  an  ordinary  measuring  scale 
along  the  diameter.  Then,  OA,  the 
projection  of  the  resultant  OP  on  the 
horizontal,  is  equal  to  6  a\,  and  OB, 
the  projection  of  the  resultant  OP  on 
the  vertical,  is  equal  to  661.  Further- 
more, if  we  write  a\  cos  x  +  b\  sin  x 
=  c\  sin  (x  +  00,  then  the  length 


FIG.  930. 


of  OP  is  6  ci  and  the  angle  FOB  is  0i.  In  Fig.  930  we  have  made  the 
construction  for  the  determination  of  ai,  bi,  c\t  and  0i  for  the  periodic 
curve  drawn  in  Fig.  89  using  the  table  of  ordinates  on  p.  184.  We  find 


OA 


134, 


di  =  -41.4,     OB  =  6bi  =  126.0,     OP  =  6 

Z  POB  =  0i  =  -18.1°; 
hence 

ai  =  —6.9,        bi  =  21.0,         Ci  =  22.3,        0i  =  —  18.1°. 

These  results  agree  very  closely  with  those  obtained  on  p.  184. 

We  may  find  a2  and  bz  by  laying  off  in  succession  the  ordinates,  each 
making  an  angle  of  60°  with  the  preceding;  we  proceed  similarly  in  finding 
the  other  coefficients.  A  separate  diagram  must  be  drawn  for  each  pair 
of  coefficients. 

More  generally,  if  we  divide  the  complete  period  into  n  equal  intervals 
of  width  2  ir/n  and  measure  the  n  ordinates,  then  (p.  177) 


202 


EMPIRICAL   FORMULAS— PERIODIC   CURVES 


CHAP.  VII 


cos  kxr  =  yQ  cos  0+3-1  cos  k  I  —  j  + 
: 3/0  sin  o +3>i  sin  k  I  — 


yn-i  cos  k 

•         i     •* 


-  flt=  "5%,  COS^ACr  =  'VnCOSO  +  Vi  COS  k\  —  I  4-    •    •    •     -\-M i  COS  k 

n 
2 

Hence,  if  we  construct  the  polygon  of  co-planar  forces  by  starting  at  a 
point  0  and  laying  off  in  succession  the  ordinates,  each  making  an  angle 

2  kir/n  with  the  preceding,  then  OA, 
the  projection  of  the  resultant  OP 
on  the  horizontal,  is  equal  to  «a*/2, 
and  OB,  the  projection  of  the  result- 
ant OP  on  the  vertical,  is  equal  to 
nbk/2,  except  when  k  =  o  or  k  =  n/2, 
when  we  get  the  values  na0,  nbo, 
nan/2,  nbn/2,  respectively.  Further- 
more, the  length  of  OP  is  n/2  (or  n) 


-3  A     -2 


FIG.  936. 


FIG.  93c. 


times  the  amplitude  Ck  and  the  angle  between  OP  and  OB  gives  the  phase 
<f>k  of  the  complete  harmonic  Ck  sin  (kx  +  0*). 

Example.     Analyze  graphically  the  periodic  curve  in  Fig.  866. 

As  in  the  example  on  p.  181,  we  shall  find  the  first  three  harmonics 
from  the  data 

o°  60°          120°    |      1 80°  240°  300° 


0.47 


1.77 


Here 


6  a3  =  y0  —  y\  +  •  • 
3  fll  =  OA  (Fig.  936) 
3  bi  =  OB  (Fig.  936) 
3  d  =  OP  (Fig.  936) 
3  02  =  OA  (Fig.  93c) 
3  62  =  OB  (Fig.  93^) 
3  c2  =  OP  (Fig.  93c) 


2.20 


3-5 


—  2.20 


o.n; 

1-95; 

3-09; 

5-35; 

6.25; 

-2.67 

-1-35 
3.00 


a, 


;    a2  =  — 


—  1.64         —0.49 


0.02. 

0-33- 
1.03. 
1.78. 
2.08, 
0.89. 
-0.45. 

1. 00, 


30°. 


02  =  -6oc 


ART.  94          MECHANICAL  EVALUATION   OF   THE   COEFFICIENTS  203 

Result: 

y  =  0.02  +  1.03  cos  x  —  0.89  cos  2  x  +  0.33  cos  3  x 

+  1.78  sin  x  —  0.45  sin  2  x 
=  0.02  +  2.08  sin  (x  +  30°)  +  sin  (2  x  -  60°)  -  0.33  sin  (3  x  -  90°). 

Note  the  close  agreement  of  this  result  with  that  obtained  by  the 
arithmetic  method  on  p.  181. 

94.  Mechanical  evaluation  of  the  coefficients.  Harmonic  analyzers. 
—  A  very  large  number  of  machines  have  been  constructed  for  finding 
the  coefficients  in  Fourier's  series  by  mechanical  means.  These  instru- 
ments are  called  harmonic  analyzers.  The  machines  have  done  useful 
work  where  a  large  number  of  curves  are  to  be  analyzed.  Among  these 
analyzers  we  may  mention  that  of  Lord  Kelvin,*  Henrici,f  Sharp,  J  Yule,§ 
Michelson  and  Stratton,|j  Boucherot,^[  Mader,**  and  Westinghouse.ff 
We  shall  briefly  describe  the  principles  upon  which  the  construction 
of  two  of  these  instruments  depend. JJ 

The  harmonic  analyzer  of  Henrici.  This  is  one  of  a  number  of  ma- 
chines which  use  an  integrating  wheel  like  that  attached  to  a  planimeter 
or  integrator  §§  to  evaluate  the  integrals  occurring  in  the  general  expres- 
sions for  the  coefficients 

i   c2r  i  r2ir  i  r2* 

a0  =  —    I      y  dx,     ak  =  -  \      y  cos  kx  dx,     b/,  =  -  I      y  sin  kx  dx 

2  7T  JQ  IT  JQ  IT  Jo 

given  on  p.  174. 

If  the  curve  in  Fig.  94/a  represents  a  complete  period  of  the  curve  to 
be  analyzed,  then  evidently 


r 


ydx  =  area  OABCDBO; 


so  that,  if  the  tracing  point  of  a  planimeter  is  allowed  to  follow  the  curve 
OABCDBO,  the  integrating  wheel  will  give  the  reading  2  TTOQ,  from  which 
ao  may  be  computed. 

*  Proc.  Roy.  Soc.,  xxvii,  1878,  p.  371;  Kelvin  and  Tait's  Natural  Philosophy. 

t  Phil.  Mag.,  xxxviii,  1894,  p.  no. 

I  Phil.  Mag.,  xxxviii,  1894,  p.  121. 

§  Phil.  Mag.,  xxxix,  1895,  p.  367;  The  Electrician,  March  22,  1895. 

H  Phil.  Mag.,  xlv,  1898,  p.  85. 

1[  Morin,  Les  Appareils  d' Integration,  1913,  p.  179. 

**  Elektrotech.  Zeit.,  xxxvi,  1909;  Phys.  Zeit.,  xi,  1910,  p.  354. 
ft  The  Electric  Journal,  xi,  1914,  p.  91. 

ft  Brief  descriptions  of  all  but  the  last  of  these  may  be  found  in  Modern  Instruments 
and  Methods  of  Calculation,  a  handbook  of  the  Napier  Tercentenary  Celebration,  1914. 
§§  For  the  principle  of  the  planimeter  and  integrator,  see  pp.  246,  250. 


204 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


Integrating  by  parts,  we  may  write 


-U": 


~rS 


ycoskxdx  =  Ij-ysinkx 


vsmkxdx  =    —  -;- 
\      kir 

Now  if  the  planimeter  carries  two 
integrating  wheels  whose  axes  make  at 
each  instant  angles  kx  and  ir/2  —  kx 
with  the  y-axis,  and  the  point  of  inter- 
section of  these  axes  is  capable  of 
moving  parallel  to  the  y-axis,  then  as 
the  tracer  point  passes  around  the 
boundary  OABCDBO,  these  wheels  give  readings  proportional  to 

I  sin  kx  dy     and       /  sin  ( kx\  dy  =  I  cos  kx  dy, 

from  which  the  values  of  a&  and  bk  can  be  found. 

In  one  form  of  the  instrument  the  curve  is  drawn  on  a  horizontal 
cylinder  with  the  ;y-axis  as  one  of  the  elements.  A  mechanism  is  attached 
to  a  carriage  which  moves  along  a  rail  parallel  to  the  axis,  by  means  of 
which  a  tracer  point  follows  the  curve  while  the  cylinder  rotates;  the 
mechanism  allows  the  axes  of  the  integrating  wheels  to  be  turned  through 
an  angle  kx  while  the  cylinder  ro- 
tates through  an  angle  x.  Coradi, 
the  Swiss  manufacturer,  has  per- 
fected the  instrument  so  that  sever- 
al pairs  of  coefficients  may  be  read 
with  a  single  tracing  of  the  curve. 


360°  X 


FIG.  946. 


FIG.  94C. 


The    Westinghouse   harmonic   analyzer.— -This   machine,    constructed 
by  the  Westinghouse  Electric  and  Mfg.  Co.,  is  particularly  useful  in 


ART.  94          MECHANICAL  EVALUATION  OF  THE   COEFFICIENTS 


205 


analyzing  the  alternating  voltage  and  current  curves  represented  by  a 
polar  or  circular  oscillogram. 

Fig.  946  gives  one  period  of  a  periodic  curve  drawn  on  rectangular 
coordinate  paper.  In  Fig.  94*;,  the  same  curve  is  represented  on  polar 
coordinate  paper.  This  is  done  by  constructing  a  circle  of  any  convenient 
radius,  called  the  zero  line  or  reference  circle  and  locating  any  point  P 
by  the  angle  6  =  x  and  the  radial  distance  r  =  y  from  the  zero  line.  Thus 
the  points  marked  P,  A,  and  B  in  Figs.  946  and  94C  are  corresponding 
points.  If  only  the  odd  harmonics  are  present,  the  second  half-period 
of  the  curve  in  Fig.  946  will  be  a  repetition  below  the  re-axis  of  the  first 
half-period;  in  this  case,  the  diameters  at  all  angles  of  the  curve  in  Fig.  946 
will  be  equal,  and  equal  to  the  diameter  of  the  reference  circle.  The  re- 
lation between  r  and  -6, 

r  =  /(0)  =  ai  cos  6  +  a2  cos  2  6  +  •  •  •   +  ak  cos  kd  +  •  •  • 
-f  bi  sin  6  -f  bz  sin  2  d  +  •  •  •   +  bk  sin  kO  +  •  •  •  , 

is  the  function  to  be  analyzed.     This  is  done  as  follows. 

The  circular  record  of  the  periodic  curve,  drawn  by  hand  from  the 
rectangular  record  or  directly  by  the  circular  oscillograph,*  is  transferred 


FIG.  94</. 

to  a  card  of  bristol  board  and  a  template  is  prepared  by  cutting  around 
the  curve.  In  the  initial  position  the  template  M  (Fig.  94^)  is  secured 
on  a  turntable  T  so  that  the  axis  6  =  o  lies  under  the  transverse  cross-bar 
B.  The  turntable  is  set  on  a  carriage  D  which  slides  on  the  rails  L.  The 

*  The  Electric  Journal,  xi,  1914,  p.  262. 


2O6 


EMPIRICAL  FORMULAS  —  PERIODIC   CURVES 


CHAP.  VII 


carriage  is  given  an  oscillatory  motion  by  the  motion  of  a  crank-pin  P 
(Figs.  94«,  94/)  attached  to  a  rotating  gear  G  and  sliding  in  a  transverse 
slot  5  on  the  bottom  of  the  carriage.  The  carriage  thus  has  a  simple 
harmonic  motion  whose  amplitude  is  the  crank-pin  radius  R.  By  means 
of  a  crank  and  a  simple  arrangement  of  gears,  the  carriage  makes  k  com- 
plete oscillations  while  the  template  makes  one  revolution,  when  deter- 
mining the  kth  harmonic. 


FIG.  942. 


The  cross-bar  B  is  attached  to  the  oscillating  carriage;  this  bar  carries 
a  pin  C  held  in  contact  with  the  edge  of  the  template  by  means  of  springs, 
so  that  the  bar  has  a  transverse  motion  as  the  template  revolves.  Re- 
ferred to  a  pair  of  axes  xx  and  yy,  the  motion  of  the  end  of  the  bar, 
Q(x,y),  may  be  said  to  consist  of  two  components,  viz.,  the  transverse 
motion  of  the  bar,  x  =  r  =  /(0),  the  function  to  be  analyzed,  and  the 
simple  harmonic  motion  of  the  carriage, 

(i)  y  =  R  sin  k6     or     (2)  y  =  R  sin  Ike  -  -}  =  -R  cos  k8, 

according  as  the  carriage  is  started  with  the  slot  S  in  the  dotted  position 
of  Fig.  94^  or  of  Fig.  94/.  A  planimeter  is  attached  with  its  tracing  point 
at  Q.  This  point  then  describes  compound  Lissajous  figures  whose  areas 
A\  and  Az  may  be  read  from  the  integrating  wheel  of  the  planimeter. 


Now  from  (i),  •— 
at) 


dy 
Rk  cos  kO  and  from  (2)  -~  =  Rk  sin  kd,  hence 


Ai  =    fxdy  =    I    ^  rRk  cos  kd  d9  =  Rk  I  'rcoskedd  =  Rkirak, 
i  Jo  Jo  Jo 

rr       '  1     ("2-r  /»2)r 

x  dy  =    /      rRk  sin  kO  d8  =  Rk   I      r  sii 
Jo  Jo 

using  the  formulas  for  ak  and  bk  on  p.  174. 


=  Rkwbk, 


Therefore 


, 
k 


Rkir' 


Gears  are  provided  to  analyze  for  all  even  and  odd  harmonics  from  I 
to  50,  and  the  shifting  of  the  gears  is  a  very  simple  matter. 


EXERCISES 


207 


EXERCISES. 

1.  Sketch  the  periodic  curves 

y  =  2  cos  x;     y  =  cos  2  x;    y  =  2  cos  x  +  cos  2  x.1 

2.  Sketch  the  periodic  curves 

y  =  i  +  sin  x;    y  =  —  f  sin  2  x;     y  =  |  sin  3  x; 
y  =  i  +  sin  x  —  ^  sin  2  x  +  3  sin  3  x. 

3.  Sketch  the  periodic  curves 

y  =  2sin(x  -  40.5°);       y  =  sin  (2  x  +  72.3°);       y  =  f  sin  (3  x  -  90°); 
y  =  2  sin  (x  —  40.5°)  +  sin  (2  x  +  72.3°)  +  £  sin  (3  x  —  90°). 

4.  Sketch  the  periodic  curve 

y  =.cos  x  +  0.4  cos  3  x  +  0.5  sin  x  —  0.5  sin  3  x. 

5.  Sketch  the  periodic  curve 

y  =  cos  x  +  0.4  cos  3  x  —  0.2  cos  5  x  +  0.5  sin  x  —  0.5  sin  3  x  —  0.3  sin  5  x. 

6.  By  use  of  the  formulas  on  p.  177  and  the  direct  method  illustrated  on  p.  179, 
determine  the  coefficients  of  the  third  and  fourth  harmonics  of  the  periodic  curve  in 
Fig.  89;  use  the  table  of  ordinates  on  p.  179. 

7.  Determine  the  first  three  harmonics  of  the  periodic  curve  given  by  the  following 
data;  use  the  computing  form  on  p.  180. 

x  o°        |      60°  120°  1 80°     I      240°     I      300° 


y  -0.85    |      0.95  0.72  2.75      |     -1.37    |    -2.20 

8.    Determine  the  first  six  harmonics  of  the  periodic  curve  given  by  the  following 
data;  use  the  computing  form  on  p.  183. 


-1*1  -39 


60°  I  90° 


120 
22 


150 
22 


180°  I  210° 


24011  270°  I  300°  I  330° 


15    I  - 


-391  -I 

9.  Determine  the  first  twelve  harmonics  of  the  periodic  curve  given  by  the  following 
data;  use  the  computing  form  on  p.  185.  (The  curve  is  a  graphical  representation  of  the 
diurnal  variation  of  the  atmospheric  electric  potential  gradient  at  Edinburgh  during  the 
year  1912.) 


15° 

30° 

45° 

60° 

-30 

-39 

-41 

-39 

90       105^ 


1  80° 


210 
IO 


225 
16 


240° 


-321   -8  |    II 
270°  I  285 


18 


135 


24 
30Q°  I  315° 


15 


J4S1 

-7 


10.  Devise  computing  forms  for  determining  the  even  and  odd  harmonic  coefficients 
using  8  and  16  ordinates  respectively. 

11.  Determine  the  odd  harmonics  up  to  the  fifth  for  the  symmetric  periodic  curve 
given  by  the  following  data;  use  the  computing  form  on  p.  187. 

30°     I     60°     I     90°     I     120 


554 


676     |     660     |     940     I    1004 

12.    Determine  the  odd  harmonics  up  to  the  fifth  for  the  symmetric  periodic  curve 
from  which  the  following  measurements  were  taken;  use  the  computing  form  on  p.  187. 

o°      I     30°  60°     I     90°     I     120°     _i5p°_ 

o  4 


9-5 


208 


EMPIRICAL  FORMULAS— PERIODIC   CURVES 


CHAP.  VII 


13.  Determine  the  odd  harmonics  up  to  the  eleventh  for  the  symmetric  periodic 
curve  from  which  the  following  measurements  were  taken;  use  the  computing  form  on 
p.  190. 


33 


45 
52 


60° 
60 


75' 


90 
27 


30 


120° 

135° 

150° 

15 

18 

6 

14.  Determine  the  odd  harmonics  up  to  the  seventeenth  for  the  symmetric  periodic 
curve  from  which  the  following  measurements  were  taken;  use  the  computing  form  on 
p.  192. 


X 

o° 

10° 

y 

0 

5 

o° 

A 

0° 
5~~ 

0° 
0 

14 

50° 

60° 

70° 

80° 

90° 

100° 

110° 

>I 
13 
4 

2 
0° 

I 

15 

2 
0° 

7 
16 

3 
o3 

0 

17 

29 

0° 

33 

42 

44 

45 

30 

31 

29 

15.  Determine  the  first  three  harmonics  for  the  periodic  curve  from  which  the  fol- 
lowing measurements  were  taken;  use  the  method  of  selected  ordinates  in  Art.  91; 
assume  that  all  higher  harmonics  are  absent. 


o" 
10.0 


5-o 


_45!_ 
5-3 

60° 

90° 

120° 

135° 

150° 

1  80° 

7-2 

6.0 

y 

-6.8 
300 

—  10.9 

I  315 

" 

8.9 

3. 

IO.O 
JO 

-17.3  1  -4.7 

210°     225°     240° 


10.7 1-3.4 


-25-9 


16    Determine  the  first  three  harmonics  for  the  periodic  curve  drawn,  in  Fig.  86b; 
use  the  method  of  selected  ordinates  in  Art.  91. 

17.  Determine  the  first  six  harmonics  for  the  periodic  curve  drawn  in  Fig.  89;  use 
the  method  of  selected  ordinates    in  Art.  91;  assume  that  all  higher  harmonics  are 
absent. 

1 8.  Determine  the  first  and  third  harmonics  for  the  symmetric  periodic  curve  given 
by  the  following  data;  use  the  method  of  selected  ordinates  in  Art.  92;  assume  that  all 
higher  harmonics  are  absent 


60° 


66.5 


22.4 


14.9 


19.  Assuming  that  the  harmonics  higher  than  the  fifth  are  negligible,  determine  the 
odd  harmonics  of  the  symmetric  periodic  curve  from  which  the  following  measurements 
were  taken;  use  the  method  of  selected  ordinates  in  Art.  92. 


* 

X 

0° 

30° 

60° 

90° 

120° 

150° 

3 

0° 

1 
I 

( 

8° 

3 

^6^ 
719 

676 

54° 

660 
702 

940 
90° 

IOO 

1  08° 

4 

12 

5 
6° 

54 
J44l 
639 

y 

o 

470 

678 

940 

1086 

920 

I62C 


375 


20.  Use  the  method  of  selected  ordinates  in  Art.  92  to  determine  the  ninth  harmonic 
of  the  curve  given  by  the  table  in  Ex.  14. 

21.  Analyze  graphically  the  curve  in  Ex.  7, 


CHAPTER  VIII. 
INTERPOLATION. 

95.  Graphical  Interpolation.  —  Having  found  the  empirical  formula 
connecting  two  measured  quantities  we  may  use  this  in  the  process  of 
interpolation,  i.e.,  in  computing  the  value  of  one  of  the  quantities  when 
the  other  is  given  within  the  range  of  values  used  in  the  determination  of 
the  formula.  It  is  the  purpose  of  this  chapter  to  give  some  methods 
whereby  interpolation  may  be  performed  when  the  empirical  formula  is 
inconvenient  for  computation  or  when  such  a  formula  cannot  be  found. 

Let  the  following  table  represent  a  set  of  corresponding  values  of  two 
quantities 

X0 


where  y  is  a  known  or  an  unknown  function  of  x.  Our  problem  is  to  find 
the  value  of  y  =  yk  for  a  value  of  x  =  Xk  between  x0  and  xn. 

A  simple  graphical  method  consists  in  plotting  the  values  of  x  and  y 
as  coordinates,  drawing  a  smooth  curve  through  or  very  near  the  plotted 
points,  and  measuring  the  ordinate  y*  of  the  curve  for  the  abscissa  x*. 
The  value  of  yk  thus  obtained  may  be  sufficiently  accurate  for  the  purpose 
in  hand.  Thus  from  the  curve  in  Fig.  726,  we  read  t  =  10,  A  =  77.0,  and 
/  =  30,  A  =  45.0.  If  we  use  the  empirical  formula  derived  on  p.  133, 
A  =  100.1  e-o-0265',  or  log^4  =  2.0005  —  0.0115 /, 

we  compute  t  =  10,  A  =  76.8  and  /  =  30,  A  =  45.2.  By  comparison 
with  the  table  on  p.  132  we  note  that  the  measured  values  of  A  for  /  =  10 
and  /  =  30  agree  about  as  closely  with  the  computed  values  as  the  neigh- 
boring observed  values  agree  with  their  corresponding  computed  values. 
Here,  the  last  significant  figures  in  the  values  of  A  were  used  in  construct- 
ing the  plot. 

On  the  other  hand,  in  Fig.  7ic,  we  read  v  =  40,  p  =  10.00,  whereas 
the  empirical  formula  on  p.  131  gives  v  =  40,  p  =  9.4.2.  The  residual 
is  0.58,  much  larger  than  the  residuals  in  the  table  on  p.  130  for  neighbor- 
ing values  of  v.  Here,  the  plot  was  constructed  without  using  the  last 
significant  figures  in  the  values  of  the  quantities.  It  is  of  no  advantage 
to  construct  a  larger  plot  since  the  curve  between  plotted  points  is  all 
the  more  Indefinite. 

209 


2IO 


INTERPOLATION 


CHAP.  VIII 


For  most  problems  the  arithmetic  or  algebraic  methods  to  be  explained 
in  the  following  sections  give  much  bettei  results. 

96.  Successive  differences  and  the  construction  of  tables.  —  Given  a 
series  of  equidistant  values  of  x  and  their  corresponding  values  of  y, 


Xi 


x0 


Xn 

x0  +  nh 


we  define  the  various  orders  of  differences  of  y  as  follows: 

1st  difference  =  A1:  a0  =  y\  —  y0,  a\  =  y%  —  yit  .  .  .  ,  an-\=  yn 
2d  difference  =  A2:  b0  =  di  —  a0,  bi  =  a<>  —  alf  .  .  .  ,  &n_2=  an 
3d  difference  =  A3:  c0  =  b\  —  bo,  c\  =  b2  —  bi,  .  .  .  ,  cn-z=  b 


kth  difference  =  A*  :  ko  =  ji  —  j0f  k\  =  j«  —  ji,  .  .  .  . 
These  may  be  tabulated  as  follows: 

x                                      y            A1            AJ           A3               A«                   ...                       A*  ... 

XQ  =  X0 

^ 

Xi   ==  XQ  ~i~  ft 

>i                     b0 

a\                     Co 

Xt  =  X0  +  2h 

y%                 bi                   do 

CL%                                Cl 

x3  =  x0  +  3  h 

y3                           bz 

O3 

ko 

X^  =  x^   +  4  h 

* 

kl 

Xvr-i  =  x0  +  (n  -  i)  h 

• 

an-i 

Xn  =  X0  +  nh 

where  a  quantity  in  any  column  of  differences  is  written  between  two 
quantities  in  the  preceding  column  and  is  equal  to  the  lower  one  of  these 
minus  the  upper  one. 

We  may  apply  the  above  definitions  in  the  formation  of  the  differences 
of  y  when  y  =  /(#)  ;  thus, 

Ay  =  f(x  +  A)  -  /(*)  =  A/(*)  ;  tfy  =  A/(*  +  A)  -  A/(*)  =  A'-/(*)  ; 
etc.  E.g.,  if 

y  =  x2  -  2  x  +  2,  Ay  =  [(x  +  h)*  -  2  (x  +  A)  +  2]  -  [x2  -  2  x  +  2] 
=  2hx+  (h*  -2  A); 


=  2  A2. 

We  note  that  A2y  =  2  &2,  so  that  the  second  differences  are  constant  for 
all  values  of  x. 


A.RT.  g6  SUCCESSIVE  DIFFERENCES 

Similarly,  if  y  =  xn  where  n  is  a  positive  integer, 
Ay  =  (x 


211 


n  (n  -  i)  xn-2/*2  +  •  >•  •  , 
n  (n  -  i)  (n  -  2}  xn~3h3  + 


&»y=n(n  -  i)  (n  -  2)  .  .  .  3  -  2  •  i  A"  =  |n  hn; 

hence  the  nth  differences  of  xn,  where  n  is  a  positive  integer,  are  constant, 
and  hence  the  nth  differences  of  any  polynomial  of  the  nth  degree 


Axn 


+  Kx  +  L, 


where  n  is  a  positive  integer,  are  constant.  If  in  forming  the  differences 
of  a  function  some  order  of  differences,  say  the  nth,  becomes  approxi- 
mately constant,  then  we  may  say  that  the  function  can  be  represented 
approximately  by  a  polynomial  of  the  nth  degree,  where  n  is  a  positive 
integer. 

The  formation  of  the  differences  for  various  functions  is  illustrated 
in  the  following  tables: 


(2) 


X 

(i) 

y 

y  =  ** 

A' 

A* 

I 

i 

7 

2 

8 

12 

19 

6 

3 

27 

18 

37 

6 

4 

64 

24 

61 

6 

5 

125 

30 

91 

6 

6 

216 

36 

127 

6 

7 

343 

42 

169 

6 

8 

512 

48 

217 

9 

729 

X 

y 

A» 

A* 

5-16 

137-39 

4-03 

5-21 

141.42 

O.08 

4.II 

5.26 

145-53 

O.O8 

4.19 

5.31 

149.72 

0.08 

4-27 

5-36 

153-99 

0.08 

4-35 

5-41 

158.34 

0.08 

4-43 

5.46 

162.77 

0.08 

5.51 

167.28 

4-51 

0.09 

4.60 

5-56 

171.88 

212 


INTERPOLATION 


LHAP.  VIII 


(3)  y  = 


(4)  y 


X 

y                   A'               A»                         ,                                 * 

y                       A1 

20 

2.7144                                                                          611 

8.4856 

445 

46 

21 

2.7589                       -14                                          612 

8.4902 

43i 

46 

22 

2.8020                               —12                                                         613 

8.4948 

419 

46 

23 

2.8439                               -13                                                         614 

8.4994 

406 

46 

24 

2.8845                               -II                                                         615 

8.5040 

395 

46 

25 

2.9240                                                                          616 

8.5086 

(5)   Train-resistance                                           (6)  Speed  of  a  vessel 

V                    R                                                           V 

speed  in       resist,  in  Ibs.          A1                A2                 speed  in 
mi.  per  hr.         per  ton                                                   knots  per  hr. 

horse-  power 

A1               AJ 

A' 

20               5.5                                               8 

I,OOO 

3-6 

400 

40                     9.1                                      2.2                        9 

1,400 

IOO 

5-8 

500 

0 

60                  14.9                                     2.1                      10 

1,900 

IOO 

7-9 

600 

50 

80                 22.8                                   2.4                    II 

2,500 

150 

10-5 

750 

50 

IOO                  33-3                                     2.2                     12 

3,250 

2OO 

12.7 

950 

50 

120                 46.0                                                             13 

4,200 

250 

] 

2OO 

IOO 

14 

5,400 

350 

] 

550 

IOO 

15 

6,950 

450 

; 

000 

50 

16 

8,950 

500 

2500 

17            11,450 

(7)  y  =  log  x                                                         (8)  y  =  log  sin  x 

x 

y                  A»                                 x 

y 

A»                A' 

A» 

500 

2.6990                                  i°   o' 

8.2419-10 

8 

e 

)69 

501 

2.6998                                 i°  10' 

8.3088-10 

-89 

9 

i 

80 

20 

502 

2.7007                                               I°20' 

8.3668-10 

-69 

9 

i 

>ii 

fi6 

503 

2.7016                                   i°3o' 

8.4179-10 

-53 

8 

i 

^58 

8 

504 

2.7024                                   i°4o' 

8.4637-10 

-45 

9 

( 

^3 

10 

505 

2.7033                                   i°  50' 

8.5050-10 

-35 

9 

\78 

506 

2.7042                                               2°     0' 

8.5428-10 

In  the  above  tables  we  note  the  following: 
In  (i),  y  =  x3  and  A3  is  constant. 

In  (2),  y  =  x3  and  A2  is  constant  since  we  have  carried  the  work  to  two 
decimal  places  and  A3  does  not  sensibly  affect  the  second  decimal  place. 


AXT.  96  SUCCESSIVE   DIFFERENCES  213 

If  the  computation  had  been  carried  to  six  decimal  places,  A2  would  not 
be  constant  but  A3  would  be. 

In  (3),  A2  is  approximately  constant,  so  that  if  we  desire  to  work  to 
four  decimal  places,  \/x  could  be  represented  by  a  polynomial  of  the 
second  degree  within  the  given  range  of  values  of  x. 

In  (4),  A1  is  approximately  constant  so  that  \^x  could  be  represented 
by  an  equivalent  polynomial  of  the  first  degree. 

In  (5)  and  (6),  A2  and  A3  are  approximately  constant,  so  that  R  may 
be  approximately  represented  by  a  polynomial  of  the  second  degree  in  V, 
and  /  by  a  polynomial  of  the  third  degree  in  V. 

In  (7),  log  x  may  be  approximately  represented  by  a  polynomial  of  the 
first  degree,  and  in  (8),  log  sin  x  by  a  polynomial  of  the  third  degree 
within  the  given  range  of  values  of  x. 

In  general,  it  is  evident  that  we  may  stop  the  process  of  finding  suc- 
cessive differences  much  sooner  the  smaller  the  number  of  digits  required 
and  the  smaller  the  constant  interval  h.  We  should  stop  immediately  if 
the  differences  become  irregular. 

The  formation  of  differences  is  often  valuable  where  a  function  is  to 
be  tabulated  for  a  set  of  values  of  the  variable.  Thus,  suppose  we  wish 
to  form  a  table  for  y  =  irx2/4,  expressing  the  area  of  a  circle  in  terms  of 
the  diameter,  for  equidistant  values  of  x.  Since  we  have  a  polynomial 
of  the  second  degree,  A2;y  is  constant,  and  if  h  =  I  and  the  work  is  to  be 
carried  to  4  decimal  places,  we  need  merely  compute  the  values  of  y  for 
x  =  i,  2,  3  and  form  the  corresponding  differences;  proceeding  back- 
wards, we  repeat  the  value  of  A2y  =  1.5708,  add  this  to  Ay  =  3.9270  and 
get  5.4978,  add  this  to  7.0686  and  get  12.5664,  which  is  the  value  of  y  for 
x  =  4.  We  proceed  in  the  same  manner  to  get  the  values  of  y  for  suc- 
cessive values  of  x. 

y  =  T*»/4  A1  A1  * 


5 


0.7854  69 

2.3562 
3.I4I6  1.5708          70 

3.9270 

7.0086  1.5708  71 

54978 

12.5664  1.5708  72 

7.0686 


3739-28 


109.17 


3848.45  1-57 

110.74 
3959-19  1.57 

112.31 
4071.50  1.57 


4185.38 


113.88 


19-6350  73 

For  larger  values  of  x  where  we  wish  to  work  to  two  decimal  places 
only,  we  take  A2;y  =  1.57  and  proceed  as  above. 

Suppose  we  wish  to  tabulate  the  function  y  =  x3.  Here  A3  is  con- 
stant so  that  we  merely  compute  the  part  of  the  accompanying  table  in 
heavy  type.  Then  we  extend  the  column  for  A3  by  inserting  6's,  extend 
the  columns  for  A2  and  A1  by  simple  additions  and  subtractions,  and  thus 
determine  the  values  of  x3  for  all  integral  values  of  x. 


214 


INTERPOLATION 


CHAP.  VIII 


X 

y-s* 

A» 

A« 

A» 

—  I 

—  I 

I 

0 

0 

0 

I 

6 

I 

I 

6 

7 

6 

2 

8 

12 

19 

6 

3 

27 

18 

37 

6 

4 

64 

24 

61 

6 

5 

125 

30 

91 

6 

216 

The  same  procedure  may  be  followed  in  the  construction  of  a  table  for 
a  function  where  a  certain  order  of  differences  is  only  approximately  con- 
stant. Thus,  in  forming  table  (4)  of  cube  roots,  we  note  that  for  that 
portion  of  the  table  Ay  is  approximately  0.0046  so  that  we  can  find  the 
values  of  \/x  by  simple  additions;  we  must  check  the  work  by  direct 
computation  every  few  values  in  order  to  find  when  A2y  changes  its  value. 

97.  Newton's  interpolation  formula.  —  We  shall  now  express  the 
value  of  y  for  any  value  of  x.  From  the  definitions  of  successive  differ- 
ences we  have 

iVi  =  yo  +  a0;       y-i  =  yi  +  ai  =  (yo  +  do)  +  (do  +  b0)  =  y0  +  2  a0  -f  b0', 
y3  =  yz  +  02  =  (yo  +  2  a0  +  b0)  +  (a0  +  2  &„  +  CD)  =  y0  +  3  «o  +  3  &o  +  c0; 
y*  =  yz  +  as  =  (>  +  3  a0  +  3  60  +  c0)  +  (ao  +  3  b0  +  3  c0  +  do) 
=  >  +  4  ao  +  6  60  +  4  co  +  do] 


We  note  that  the  coefficients  are  those  of  the  binomial  expansion,  and 
this  suggests  that 

n(n  -  i)  t    ,  n(n  -  i)  (n  -  2)       ,  m 

'  >         v1/ 


where  n  is  a  positive  integer.     If  this  equation  is  true,  then,  replacing  y 
by  a,  the  first  difference,  we  may  also  write 

,     ,  n  (n  -  i)           n(n  -  i}(n-  2) 
dn  =  a0  +  woo  H r- c0  H 1- 

L£  i^ 

r..  /. 
(»• 


t  rn.(n  -  i)  (n  -  2)   |  n  (n  -  i) 


j_  /-         T^       , 
=  yo  +  (n  +  i)  do  H 


ART.  97  NEWTON'S  INTERPOLATION  FORMULA  215 

where  the  coefficients  are  again  those  of  the  binomial  expansion  with  n 
replaced  by  n  +  i.  Thus  we  have  shown  that  if  equation  (I)  is  true  for 
any  positive  integral  value  of  n,  it  is  true  for  the  next  larger  integral  value. 
But  we  have  shown  (I)  to  be  true  when  n  =  4,  therefore  it  is  true  when 
n  =  5  ;  since  it  is  true  for  n  —  5,  therefore  it  is  true  for  n  =  6  ;  etc.  Hence 
(I)  is  true  for  all  positive  integral  values  of  n. 

Now  if  some  order  of  differences,  say  the  kth  order,  is  constant,  i.e., 
&ky  =  ko,  then  y  is  a  polynomial  of  the  kth  degree  in  n,  and  equation  (I) 
may  be  written 

A  +  Bn  +  C«2  +  •  •  •  +  Ln*  =  y0  +  no*  +  n  ("  ~  T)  b0  +  .  .  . 

,  n  (n  -  i)  .  .  .  (n  -  k  +  i)  , 
~\j~  ~ko' 

The  right  member  of  this  equation  is  also  a  polynomial  of  the  kth  degree 
in  n,  and  since  these  polynomials  are  equal  for  all  positive  integral  values 
of  n  (i.e.,  for  more  than  k  values  of  «),  they  must  be  equal  for  all  values  of 
n,  integral,  fractional,  positive,  and  negative. 

Hence  if  the  kth  order  of  differences  is  constant,  we  have 


L5 

for  all  values  of  n.  This  fundamental  formula  of  interpolation  is  known 
as  Newton's  interpolation  formula.  In  this  formula,  y0  is  any  one  of  the 
tabulated  values  of  y  and  the  differences  are  those  which  occur  in  a  line 
through  y0  and  parallel  to  the  upper  side  of  the  triangle  in  the  tabular 
scheme  on  p.  210. 

Newton's  formula  is  approximately  true  for  the  more  frequent  case 
where  the  differences  of  some  order  are  approximately  constant;  all  the 
more  so  if  n  <  i.  We  can  always  arrange  to  have  n  <  i  ;  for  if  we  wish 
to  find  the  value  of  y  =  Y  for  x  =  X,  where  X  lies  between  the  tabular 
values  Xi  and  Xi+i,  we  use  Newton's  formula  with  y,-  and  the  correspond- 

ing differences  a<,  bi,  d,  .  .  .  ,  so  that  X  =  xf  +  nhandn  =  —  r—^<t.J| 

ft 

The  values  of  the  binomial  coefficients  occurring  in  the  formula  have 
been  tabulated  for  values  of  n  between  o  and  i  at  intervals  of  o.oi.f 

Let  us  now  apply  Newton's  formula  to  the  illustrative  difference- 
tables  (i)  to  (8). 

*  The  ordinary  interpolation  formula  of  proportional  parts  disregards  all  differences 
higher  than  the  first,  so  that  y  —  yo  +  nao,  where  n  —  (X  —  x0)/h.  This  simple  formula 
will  often  give  the  desired  degree  of  accuracy  if  the  interval  h  can  be  made  small  enough. 

t  See  H.  L.  Rice,  Theory  and  Method  of  Interpolation. 


216 


INTERPOLATION 


CHAP.  VIII 


(i)  To  compute  (3.4)*;  yo  =  27>  h=i,  w=  (3.4— 3)71=0.4; 
.-.     (3.4)'  =  27  +  (o.4)  (37)+  (°-4)(-°-6)  (24)  +  ($4)  (-Q-6)  (-1-6)   (fi) 

2  6 

=  39.304. 

(3)  To  compute  \/2^5;  y0  =  2.8439,  h  =  i,  n  =(23.5  -  23)71  =  0.5; 
/.     ^23.5  =  2.8439  +  \  (0.0406)  +  |  (o.ooii)  =  2.8643. 

If  we  use  the  ordinary  interpolation  formula  of  proportional  parts, 
-^23.5  =  2.8439  +  \  (0.0406)  =  2.8642,  which  would  be  correct  to 
three  decimals  only. 

(4)  To  compute  ^612.25;  3-0  =  8.4902,  h=i,  «  =  (612.25-612)71  =£; 
/.     v"6i2.25  =  8.4902  +  \  (0.0046)  =  8.4914. 

(5)  To  computed  when  7=65;  ^0  =  14-9,  A  =  20,  n  =  (65  —  6o)/2o  =  i :; 
/.     R  =  14.9  +  J  (7.9)  -  TJ\  (2.4)  =  16.7. 

(7)  To  compute  log  501. 3;  3-0  =  2.6998,  h=  i,  n=  (501. 3-501)71  =0.3; 
/.     log  501.3  =  2.6998  +  0.3  (0.0009)  =2.7001. 

(8)  To   compute    log  sin  i°  16';   3-0  =  8.3088  —  10,    h  =  10',    n  = 
(i°  1 6'  -  i°  io')/io'  =  0.6; 

.'.     log  sin  i°  16'  =  (8.3088  —  10)  +  0.6  (0.0580)  —  0.12  (  —  0.0069) 
+  0.056  (0.0016)  =  8.3445  —  I0»  correct  to  4  decimals. 

If  we  use  the  ordinary  formula  of  proportional  parts,  we  have 
log  sin  i°  16'  =  8.3088  —  10  +  0.6  (0.0580)  =  8.3436  —  10,  correct  to  2 
decimals  only. 

If  the  value  of  x  for  which  we  wish  to  determine  the  value  of  y  is  near 
the  end  of  the  table  we  may  not  have  all  the  required  differences.  To 
take  care  of  this  case  Newton's  formula  is  slightly  modified.  If  we 
invert  the  series  of  values  of  x  in  the  tabular  scheme  on  p.  210,  and  form 
the  differences,  we  have 


yn 
yn-i 


-On-l 


—  Ci 

—Co 


ART.  97  NEWTON'S  INTERPOLATION  FORMULA  2  17 

t  y\  and 

,        . 
n  (-a3) 


Starting  at  y\  and  applying  Newton's  formula,  we  get 

n  (n  —  i)  ,     .  n  (n  —  i)  (n  —  2\  , 


,  n(n-i)i       n  (n  -  i)  (n  -  2)  _ 
=  y4  -  wa3  H  ---  ^  -  62  -  ^          -  ci  +  • 

Comparing  the  result  with  the  scheme  on  p.  210,  we  note  that  the  differ- 
ences are  those  which  occur  along  a  line  parallel  to  the  lower  side  of  the 
triangle  in  that  scheme.  Here  y\  is  any  value  of  y,  and  if  X  lies  between 
#4  and  x3,  then  X  =  #4  —  nh,  and  n  =  (x*  —  X)/h. 

Example.  To  compute  ^24.8.  In  table  (3),  y4  =  2.9240,  h  =  i, 
n  =  (25  -  24.8)71  =  0.2; 

.*.   "V/24.8  =  2.9240  —  0.2  (0.0395)  H  —  '  —  -  —  (—0.0011)  =  2.9162. 

If  a  series  of  corresponding  numerical  values  of  two  quantities  are 
given,  we  may  use  Newton's  formula  for  finding  the  polynomial  which 
will  represent  this  series  of  values  exactly  or  approximately.  For  this 
purpose  we  replace  n  by  (x  —  Xo)/h. 

Thus,  in  table  (i),  h  =  i,  XQ  =  I,  n  =  x  —  I  ; 


i    /  \      7  V  —  20       V 

In  table  (5),     h  =  20,     F0  =  20,     w  =  —  —  —  =  —  -  1  ; 


=  4.1  +  0.015  v  +  0.00275  y2. 

The  values  of  .R  computed  by  this  formula  agree  quite  closely  with  those 
in  the  table. 

In  table  (6),  h  =  i,     F0  =  10,     n  =  V  -  10; 

/.     /  =  1900  +  (V  -  io)  600  +  (V  ~  IO)2(F  ~  JI)  150 

(F-io)(F-n)(F-i2) 

6 

=  -6850  +  2042  V  -  200  F2  +  8|  F3. 

The  values  of  7  computed  by  this  formula  agree  quite  closely  with  those 
in  the  table;  thus,  F  =  12  gives  /  =  3254. 

Various  formulas  of  interpolation  similar  to  Newton's  have  been  de- 
rived which  are  very  convenient  in  certain  problems.  Among  these 
may  be  mentioned  the  formulas  of  Stirling,  Gauss,  and  Bessel.* 

*  For  an  account  of  these  formulas,  see  H.  L.  Rice,  Theory  and  Practice  of  Interpola- 
tion, and  D.  Gibb,  Interpolation  and  Numerical  Integration. 


2l8 


INTERPOLATION 


CHAP.  VIII 


98.  Lagrange's  formula  of  interpolation.  —  Newton's'  formula  is 
applicable  only  when  the  values  of  x  are  equidistant.  When  this  is  not 
the  case,  we  may  use  a  formula  known  as  Lagrange's  formula.  Given 
the  following  table  of  values  of  x  and  y, 


X      \ 


a2 


C.I 


3':. 


we  are  to  find  an  expression  for  y  corresponding  to  a  value  of  x  lying  be- 
tween a\  and  an.  We  take  for  y  an  expression  of  the  (n  —  i)st  degree  in 
x  containing  n  constants,  and  determine  these  n  constants  by  requiring 
the  n  sets  of  values  of  x  and  y  to  satisfy  the  equation.  But  instead  of 


assuming  the  form  y  =  A  +  Bx  +  Cx2  +  •  • 

the  equivalent  form 

y  =  A  (x  -  a2)  (x  -  a3)  (x  -  a4) 
+  B  (x  -  ai)  (x  -  as)  (x  -  a4) 
+  C  (x  -  ai)  (x  -  02)  (*  -  a*) 


+  Nx"^,  we  may  assume 

.  .  (x  -  an} 
.  .  (x  -  an} 
..(*-  On) 


+  TV  (x  -  ai)  (x  -  02)  (x  -  a8)   .  .  .  (x  -  an-i), 

where  the  w  terms  in  the  right  member  of  the  equation  lack  the  factors 
(x  —  Ci),  (x  —  os),  •  •  •  (x  —  an}  respectively. 
Since  (a\,  yi)  is  to  satisfy  this  equation, 

y\  =  A  (ai  -  02)  (ai  -  c3)  (ai  -  a4)  .  .  .  (ai  -  a«), 

since  all  the  other  terms  contain  the  factor  (ai  —  ai)  and  therefore  vanish. 
Similarly, 

j2  =  B  (0-2  -  aO  (02  -  as)  (02  -  a4)  .  .  .  (a2  -  an), 
y3  =  C  (a8  -  ai)  (a3  -  a2)  (as  -  a4)  .  .  .  (as  -  an), 


Hence,    [ 


N  (an  -  ai)  (aw  - 


-  a3)   .  .  .  (an  -  an_i 


(ai  —  a%)  (ai 

^2 

.  .  (ai  —  an) 

-,  etc., 
.  (x-aj 

(a2  —  ai)  (flz 
and,  finally, 

(*-<fc)  (*-fls)  •  •  •  (* 

—  a»)  (a2  —  J4)  . 

.  .  (a,  -  a.) 
)  (ac  —  a3)  .  . 

"^     "yi(ai  —  j^)  (fli  —  fls)   ...  (a 
.     (*~ai> 

-i  —  an)       Z(a2—  ai 
i  (x  —  a?)  .  .  .  (x 

)  (a2  —  a3)  .  . 
-  fln-i) 

'  ((h~an} 

(an  - 


«  -  a2) 


(a«  -  a«_0 

We  note  that  in  the  term  containing  yk,  the  numerator  of  the  fraction 
lacks  the  factor  (x  —  a*)  and  the  denominator  lacks  the  corresponding 
factor  (a*  —  a*).  Lagrange's  formula  is  in  convenient  form  for  logarith- 
mic computation.  • 


ART.  99 


INVERSE   INTERPOLATION 


2I9 


Example.     In  the  table  on  p.  132  we  have 


'7 


/ 

J4 

17 

3i 

A 

68.7 

64.0 

44.0 

and  we  are  to  find  the  value  of  A  when  t 
(27-17)  (27-31)  (27-35) 
(14-17)  (14-31)  (14-35) 
14)  (27- 17)  (27-35) 


,  6 
+4 


I    .. 
' 


. 
' 


39-1 

27.     Using  Lagrange's  formula, 
(27-14)  (27-31)  (27-35) 
(17-14)  (17-31)  (17-35) 
(27-14)  (27-17)  (27-31) 


(3i -14)  (31 -17)  (31 -35) 
=  -20.5  +  35.2  +  48.0  -  13.4  =  49.3, 

which  agrees  exactly  with  the  observed  value. 
Example.     In  the  table  on  p.  157  we  have 

O.I  0.2        I        O.4 


(35 -14)  (35 -17)  (35 -3i) 


2.48 


0.8 


and  we  are  to  find  the  value  of  i  when  / 
/  =  0.2  and  t  =  0.4, 


2.66     I     2.58     |     2.00 

0.3.     Using  only  the  values 


*  =  2.66 


H^  +  ^!f^i='-33  +  '.*9-.6, 


Using  all  four  values  of  /,  i  =  2.68.     Using  the  empirical  equation 
i  =  4.94  erljat  —  2.85e~3-76'  (on  p.  159),  we  get  i  =  2.66. 
Gauss's  interpolation  formula  for  periodic  functions.  —  When  the  data 
are  periodic  we  may  find  the  empirical  equation  as  a  trigonometric  series 
by  the  method  of  Chapter  VII  and  use  this  equation  for  purposes  of  in- 
terpolation, or  we  may  use  an  equivalent  equation  given  by  Gauss: 
sin  %  (x  —  02}  sin  \  (x  —  a3)  .  .  .  sin  \  (x  —  an) 


y  = 


sin  ^ 
sin 


at  —  a2)  sin  \  (ai  —  a3) 
(x  —  aO  sin  \  (x  —  o3) 


sn      a 
sin  \  (x 


—  an) 
—  an) 


—  ai)  sin  \  (a2  -  a3)  .  .  .  sin 


-  an) 


It  is  evident  that  y  =  y\  when  x  =  a\,  y  =  Ji  when  x  =  a%,  etc.,  so  that 
the  equation  is  satisfied  by  the  corresponding  values  of  x  and  y. 
99.  Inverse  interpolation.  —  Given  the  table 


X 

#0 

Xi 

Xt 

x& 

.  .   . 

*n 

y 

y0 

yi 

yz 

y3 

.   .   . 

yn 

we  may  wish  to  find  the  value  of  x  corresponding  to  a  given  value  of  y. 
If  the  values  of  x  are  equidistant  we  may  use  Newton's  interpolation 
formula.     Here  we  know  yn,  y0,  a0,  &o,  CQ,  .  .  .  ,  and  substituting  these 
values  in  the  formula  we  have  an  equation  which  is  to  be  solved  for  n. 
If  only  the  first  order  of  differences  are  taken  into  account,  then 

yn  =  yo  +  noo,  and  n  =  — — ,  the  ordinary  formula  for  inverse  inter- 

a0 

polation  by  proportional  parts. 


220  INTERPOLATION  CHAP.  VIII 

Example.     In  table  (7),  given  log  x  =  2.7003,  to  find  x. 

'  and  x  =  x0  +  nh  =  501+0.56(1)  =501.56. 


If  only  the  first  and  second  differences  are  taken  into  account,  then 
yn  =  yo  +  na0  +  --  b0,  a  quadratic  equation  which  can  easily  be 

solved  for  «. 

Example.     In  table  (5),  given  R  =  27.3,  to  find  V. 

Here  27.3  =  22.8  +  n  (10.5)  +  *  (n  ~  ^  (2.2), 

or  i.i  w2  -f  9.4  n  —  4.5  =  o; 

hence    n  =  T5T  =  0.455  and  x  =  V0  +  nh  =  80  +  (0.455)  20  =  89.1. 

The  empirical  formula  R  =  4.62  —  0.004  ^  ~H  0.0029  V2  on  p.  149 
gives  V  =  89.1,  R  =  27.3. 

But  if  the  third  and  higher  orders  of  differences  have  to  be  taken  into 
account,  the  method  would  require  the  solution  of  equations  of  the  third 
and  higher  degrees.  In  such  cases  as  well  as  in  the  case  where  the  values 
of  x  are  not  equidistant,  we  may  use  Lagrange's  formula  and  merely  in- 
terchange x  and  y;  i.e., 


(a]  —  a2)  (a!  —  a3)   .  .  .  (a2  —  fli)  (a2  —  fit)  .  .  . 

Example.     In  table  (8),  given  log  sin  #  =  8.3850  —  10,  to  find  x. 
Using  only  the  following  values, 


log  sin  x       I    8.3088  -  10    1    8.3668  -  10 


x  70'  80' 


5.4179  —  10 


90' 


we  have 


x  =  70'    (0-0182)  (-0.0329)         go/  (0.0762)  (-0.0329) 
(-0.0580)  (-0.1091)  (0.0580)  (-0.0511) 

,  (0.0762)  (0.0182) 
°  (0.1091)  (0.0511) 

=  70'  (-0.0946)  +  80'  (0.846)  +  90'  (0.249) 
=  83.47'  =  i°  23.47'. 

We  may  also  use  a  method  of  successive  approximations  as  follows: 
From  Newton's  formula  we  write 


(n  -  i)  b0  +  \(n  -  i)  (»  -  2)  c0  +  •  •  • 

Applying  this  to  the  above  example,  and  taking  only  the  first  differences 
into  account,  we  get  as  a  first  approximation, 

n  =  y~y°  =  (8.3850  ~  10)  ~  (8.3668  -  10)  =  182  =  Q 
Co  0.0511  511 


ART.  99  INVERSE  INTERPOLATION  221 

Taking  also  second  differences  into  account  and  introducing  the  value  of 
n\  for  w  in  the  denominator,  we  get  as  a  second  approximation, 

_  y  —  y0  0.0182  _  182  _ 

***  ~  OQ  +  \  (wi  —  i)  b0  ~  0.0511  +  0.0017  ~  528  ™ 

We  may  continue  in  this  way  approximating  more  and  more  closely  to  the 
value  of  n.     In  this  example  it  will  be  unnecessary  to  carry  the  work  to 
third  differences  since  A3  is  negligible.     Hence 
n  =  0.345,     and    x  =  x0  +  nh  =  i°  20'  +  (0.345)  (10')  =  i°  23.45'. 
We  may  check  this  by  direct  interpolation.     Here 

yo  =  8.3668  —  10,     h  =  10',  and     n  =  0.345; 
hence, 
y  =  8.3668  -  10  +  0.345  (0.0511)  -  0.113  (-0.0053)  =  8.3850-10. 

Example.     Find   the  real   root  of  the  equation  Xs  +  5  x  —  1=0. 
We  form  a  table  of  differences  of  the  function  y  —  x*  -J-  5  x  —  I. 


-19 

-7 


-6 
o 
6 

12 


The  root  lies  between  x  =  o  and  x  =  I,  and  we  are  to  find  the  value  of  x 
when  y  =  o.     Using  the  method  of  successive  approximations  we  have 

o  4-  i       i 


y  — 


y  -  yo 


6  +  |(l_l)6  =  f  =  o.2857, 


fl<>  +  1  («2  -   I)  bo  +  I  («2  ~   I)  (**  ~  2)  C0         6—^+1*         249 


%  =   0.1968, 


0.1985. 


4-  i  («s  -  i)  («3  -  2)  c0      6  -  2.4096  +  1.4483 


nh  =  0.1985. 


5-0387 
Hence, 
From  the  table 


we  note  that  x  =  0.1984  is  the  root  correct  to  4  decimals. 


X 

0.1985 

0.19845 

0.1984 

y 

0.00032 

0.00006 

—  0.00019 

222 


INTERPOLATION 


CHAP.  VIII 


EXERCISES 

1.  Tabulate  the  values  and  differences  of  the  following  functions;  h  is  the  common 
interval. 

(a)  x2,  from  x  =  5  to  x  ='  12  when  h  =  i ;  and  from  *  =  3  to  x  =3.1  when  h  -  o.oi. 

(b)  V#,  from  x  =  i  to  x  =  10,  when  h  =  i,  and  from  x  =  563  to  *  =  570  when 
h  =  i. 

(c)  -,  from  x  =  60  to  x  =  70  when  h  =  i,  and  from  x  =  260  to  x  =  262  when 
h  =  0.2. 

(d)  —r-  (volume  of  a  sphere),  from  D  =  I  to  D  =  1.8  when  A  =  o.l. 

(e)  log  x,  to  4  decimals,  from  *  =  356  to  x  =  362  when  h  =  i . 

(f)  tan  x,  to  4  decimals,  from  x  =  32°  to  x  =  33°  when  h  =  10'. 

(g)  log  cos  x,  to  4  decimals,  from  x  =  88°  10'  to  *  =  89°  20'  when  h  =  10'. 
(h)  <?,  to  4  decimals,  from  x  =  0.8  to  x  =  0.9  when  A  =  o.oi. 

(i)  \  (a  —  sin  a),  to  4  decimals  (area  of  a  segment  of  a  circle  subtending  a  central 
angle  a,  in  radians)  from  a.  =  25°  to  a  =  32°  when  h  =  1°. 

2.  Tabulate  the  differences  for  the  following  experimental  results  and  indicate  for 
each  case  the  degree  of  the  polynomial  that  would  best  express  the  relation  between  the 
variables. 

(a)  5  =  stress  in  Ibs.  per  sq.  in.  in  steel  wire  used  for  winding  guns,  E  =  elongation 
in  inches  per  inch. 

S  \  10,000       20,000    |    30,000 


40,000 


50,000        60,000 


70,000 


80,000 


0.00019     0.00057       0.00094       0.00134       0.00173   |   0.00216       0.00256      0.00297 
(b)  Q  =  cu.  ft.  of  water  per  sec.  flowing  over  a  Thomson  gauge  notch  ;H  =  ft.  of  head. 


H 

1.2 

1.4 

1.6 

Q 

4.2 

6.1 

8-5 

14.9 

(c)  P/a  =  load  in  Ibs.  per  sq.  in.  which  causes  the  failure  of  long  wrought-iron 
columns  with  round  ends,  l/r  =  ratio  of  length  of  column  to  least  radius  of  gyration  of 
its  cross-section. 


l/r 
P/a 


140 

18: 

12,800 

7500 

220 


260 


5000 


3800 


300 


2800 


340 


380 


1700 


(d)  e  =  volts,  p  =  kilowatts  in  a  core-loss  curve  for  an  electric  motor. 

!0          I  140          1 


e 

40 

60 

80 

100 

P 

0.63 

1.36 

2.18 

3-oo 

1 60 


3-93 


6.22 


8-59 


(e)  A  —  amplitude  of  vibration  in  inches  of  a  long  pendulum,  t  =  time  in  mio, 
since  it  was  set  swinging. 

t       |         o  i  2  3  4         |         s  6 


A     |        10  4.97  2.47  1.22  0.61       |      0.30  0.14 

(/)   V  =  potential  difference  in  volts,  A  =  current  in  amperes  in  an  electric  circuit. 


A 

2.97 

V 

65.0 

61.0 


4-97 


5-97 


58-25 


56.25 


6.97 


55- 


(g) 
x 


y    i     6.42 


8.50 


11.03 


14.03 


17-53 


21.55 


EXERCISES  223 

(A) 


x   I      o       I     0.3     I     0.6     I     0.9 
~y~\    3.00    I     1.89    I    1.27    I    0.88 


1.5     I     1.8     I     2.1     I     2.4     I    2.7 
0.46    I    0.33    I    0.25    I    0.18    I  0.05 


0.63 

3.  By  the  method  of  differences  explained  in  Art.  96,  extend  the  tabulation  of  the 
functions  in  Exs.  I  a,  b,  d,  e,  h,  i,  for  several  values  of  the  variables  beyond  the  range 
of  values  for  which  the  tables  were  constructed. 

4.  Apply  Newton's  interpolation  formula  to  the  tables  in  Ex.  I. 

(a)  In  Ex.  1  a,  find  xy  when  x  =  7.3  and  x  =  3 .056. 

(b)  In  Ex.  i  b,  find  V*  when  x  =  566.2. 

(c)  In  Ex.  i  d,  find  irD3/6  when  D  =  1.452. 

(d)  In  Ex.  i  e,  find  log  x  when  x  =  361.4. 

(e)  In  Ex.  i  g,  find  log  cos  x  when  x  =  88°  43'. 

5.  Apply  Newton's  interpolation  formula  to  the  tables  in  Ex.  2. 

(a)  In  Ex.  2  a,  find  E  when  5  =  42,000. 

(b)  In  Ex.  2  b,  find'Q  when  H  =  1.7,  and  compare  with  the  value  given  by  the  em- 
pirical formula  Q  =  2.672  H2-48. 

(c)  In  Ex.  2  c,  find  P/a  when  l/r  =  327,  and  compare  with  the  value  given  by  the 
empirical  formula  P/a  =  417,000,000  (l/r)2-1 

(d)  In  Ex.  2  f,  find  V  when  A  =  4.07. 

(e)  In  Ex.  2  g,  find  y  when  x  =  6. 

(f)  In  Ex.  2  h,  find  y  when  x  =  1.3  and  x  =  2.46. 

6.  In  the  following  table  (taken  from  p.  129) 


288  293 


313 


45-8 


333 


55-2 


•*      I     35-2      |     37-2 

S  is  the  number  of  grams  of  anhydrous  ammonium  chloride  which  dissolved  in  100  grams 
of  water  makes  a  saturated  solution  of  6°  absolute  temperature.  Use  Lagrange's  formula 
of  interpolation  to  find  S  when  8  =  300°,  using  (i)  only  two  values  of  0,  (2)  three  values 
of  9,  (3)  all  four  values  of  6.  Compare  the  results  with  the  value  given  by  the  empirical 
formula  5  =  0.000000882  03-09. 

7.    In  the  following  table  (taken  from  p.  141) 

I        I       2        |       4       I       8 


120      |       94       I       75  62 

i  is  the  current  and  V  is  the  voltage  consumed  by  a  magnetite  arc.  Use  Lagrange's 
formula  to  find  V  when  i  =  3,  and  compare  the  result  with  the  value  given  by  the 
empirical  formula  V  —  30.4  +  90.4  i-0-607. 

8.  Use  the  methods  of  inverse  interpolation  (Art.  99)  in  the  following: 

(a)  In  Ex.  i  a,  find  x  when  xz  —  39  and  when  x2  =  9.34. 

(b)  In  Ex.  i  e,  find  x  when  log  x  =  2.5542. 

(c)  In  Ex.  i  g,  find  x  when  log  cos  x  =  8.3946  —  10. 

(d)  In  Ex.  2  a,  find  5  when  E  =  0.00192. 

(e)  In  Ex.  2  c,  find  l/r  when  P/a  =  4000. 

(f)  In  Ex.  2  g,  find  x  when  y  =  15.25. 

9.  Approximate  to  the  real  roots  of  the  equations: 

(a)  x3  —  2  x  +  3  =  o. 

(b)  x*  -  4  x  +  2  =  o. 

(c)  e  +  x2  -  4  =  o. 

(d)  10  log  x  —  x  —  2  =  o. 

(e)  sin*  +  x2  —  1.5  =  o. 


CHAPTER  IX. 
APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION. 

100.  The  necessity  for  approximate  methods.  —  In  a  large  number 
of  engineering  problems  it  is  necessary  to  determine  the  value  of  the 

definite  integral,  /    f(x)  dx.     Geometrically,  this  integral  represents  the 

area  bounded  by  the  curve  y  =  /(*),  the  re-axis,  and  the  ordinates  x  =  a 
and  x  =  b.  Physically,  it  may  represent  the  work  done  by  an  engine, 
the  velocity  acquired  by  a  moving  body,  the  pressure  on  an  immersed 
surface,  etc.  If  f(x)  is  analytically  known,  the  above  integral  may  be 
evaluated  by  the  methods  of  the  Integral  Calculus.  But  if  we  merely 
know  a  set  of  values  of  f(x}  for  various  values  of  x,  or  if  the  curve  is  drawn 
mechanically,  e.g.,  an  indicator  diagram  or  oscillograph,  or  even  where  the 
function  is  analytically  known  but  the  integration  cannot  be  performed 
by  the  elementary  methods  of  the  Integral  Calculus  —  in  all  these  cases, 
the  integral  must  be  evaluated  by  approximate  methods  —  numerical, 
graphical,  or  mechanical.  The  planimeter  is  ordinarily  used  in  measuring 
the  area  enclosed  by  an  indicator  diagram  and  in  certain  problems  in 
Naval  Architecture;  such  approximations  often  have  the  desired  degree 
of  accuracy.  Where  a  higher  degree  of  accuracy  is  required  or  where  a 
planimeter  is  not  available  numerical  methods  must  be  used. 

In  certain  problems  it  becomes  necessary  to  determine  the  value  of 

dy 
the  derivative,  -7- .     Geometrically,  this  represents  the  slope  of  the  curve 

y  =  /(*)  at  anY  point.  Physically,  it  arises  in  problems  in  which  the 
velocity  and  acceleration  are  to  be  found  when  the  distance  is  given  as  a 
function  of  the  time,  in  problems  involving  maximum  and  minimum 
values  and  rates  .of  change  of  various  physical  quantities,  etc.  To 
evaluate  the  derivative  we  may  use  the  methods  of  the  Differential  Cal- 
culus if  the  function  is  analytically  known.  Otherwise  we  are  forced  to 
use  approximate  methods  —  numerical,  graphical,  or  mechanical. 

It  is  our  purpose,  in  the  following  sections,  to  develop  some  of  the 
numerical,  graphical,  and  mechanical  methods  used  in  approximate  in- 
tegration and  differentiation. 

101.  Rectangular,  Trapezoidal,  Simpson's,  and  Durand's  rules.  — 
Suppose  we  wish  to  find  the  approximate  area  bounded  by  the  curve 
V  —  /(*)»  tne  *-axis,  and  the  ordinates  x  =  x0  and  x  =  xn  (Fig.  101). 

224 


ART.  ioi         TRAPEZOIDAL,   SIMPSON'S,  AND   DURAND'S  RULES 


225 


We  divide  the  interval  from  x  =  XQ  to  x  =  xn  into  n  equal  intervals  of 
width  h,  and  measure  the  (n  +  i)  ordinates  y0,  yi,  y*,  .  .  .  ,  yn-i,  yn. 

(i)  Rectangular  rule.  —  If,  starting  at  PQ,  we  draw  segments  parallel 
to  the  x-axis  through  the  points  P0,  PI,  PZ,  -  -  •  ,  Pn-\,  the  area  enclosed 
by  the  rectangles  thus  formed  is  given  by 


FIG.  ioi. 

If,  starting  at  Pn,  we  draw  segments  parallel  to  the  x-axis  through  the 
points  Pn,  Pn-i,  .  .  .  ,  P2,  PI,  the  area  enclosed  by  the  rectangles  thus 
formed  is  given  by 

AR'  =  h  CV!  +  y2  +  y,  +  •  •  •   +  yn). 

It  is  evident  that  the  smaller  the  interval  h,  the  better  the  approxima- 
tion to  the  required  area. 

(2)  Trapezoidal  rule.  —  If  the  chords  P0Pi,  PiPz,  .  .  .  ,  Pn-iPn  are 
drawn,  then  the  area  enclosed  by  the  trapezoids  thus  formed  is 


*Mi  Cvo  +  yn) 

This  expression  for  the  area  is  the  average  of  the  two  expressions  given 
by  the  rectangular  rules.  It  is  evident  that  the  smaller  the  interval  h 
and  the  flatter  the  curve,  the  better  the  approximation  to  the  required 
area.  If  the  curve  is  steep  at  either  end  or  anywhere  within  the  interval, 
the  rule  may  be  modified  by  subdividing  the  smaller  interval  into  2 
or  4  parts;  thus,  subdividing  the  steep  interval  between  xn-\  and  xn  in 
Fig.  ioi 

into  2  parts-.  AT  =  '  ^  +  y^  -         -   -  *  /y-  +  y^  -  *  <y" 


i±&)*4(fc±») 


226       APPROXIMATE   INTEGRATION   AND   DIFFERENTIATION         CHAP.  IX 

into  4  parts:  AT  =  h  fy°      yi j  +  •••+-  ly*~l      yk j  +  - 
ym  +  yi\  ,  h  /yi  +  yn\ 

2  ;+4v  2  ;• 

(3)  Simpson's  rule.  —  Let  us  pass  arcs  of  parabolas  through  the  points 
PoPiPz,  PiPzPi,  .  .  .  ,  Pn-zPn-iPn-  Let  the  equation  of  the  parabola 
through  PoPiPz  be  y  =  ax2  +  foe  +  c.  Then  the  area  bounded  by  the 
parabola,  the  x-axis,  and  the  ordinates  x  =  x0  and  x  =  x-t  is 

ax3  ,  bx2  ,       ha,,  ,6, 

OX  +  C)  aX  = H 1-  CX      =  -  (X2    —  XQ  )  H (X2    —  Xn  ) 

32  «»       3  2V 

)  +  6cl. 


Now,      ^o  =  flx02  +  bx0  -\-  c,     yz  =  axz2  +  bx2  -\-  c,     h  = 


( 
a  f 


=  ax-          xi      c  =  a    —  —  ,,  c> 

and  we  may  easily  verify  that 

A  =  i  h  (y0  +  4^!  +  3k)  • 

If  we  have  an  even  number  of  intervals  and  apply  this  formula  to  the 
successive  areas  under  the  parabolic  arcs,  we  get 


(yo  +  4  yi  +  2  ^2  +  4  Js  +  2  y4  +   •  •  •   +2  ;yn_2  +  4  y^j  +  yB) 

•  +y«-0 


To  apply  Simpson's  rule  we  must  divide  the  interval  into  an  even 
number  of  parts,  and  the  required  area  is  approximately  equal  to  the 
sum  of  the  extreme  ordinates,  plus  four  times  the  sum  of  the  ordinates 
with  odd  subscripts,  plus  twice  the  sum  of  the  ordinates  with  even  sub- 
"scripts,  all  multiplied  by  one-third  the  common  distance  between  the 
ordinates. 

(4)  Durand's  rule.*  —  If  we  have  an  even  number  of  parts  and  apply 
Simpson's  rule  to  the  interval  from  Xi  to  xn-\  and  the  Trapezoidal  rule  to 
the  end  intervals, 


+  !  yn-a  +  J  y«-2  +  I  yn-i)  +  (I  yn-i  +  *  y.)]. 
Applying  Simpson's  rule  to  the  entire  interval  from  x0  to  xn, 

4=A[$yo+  Jyi  +  f  y,  +  $y,  +  •  -  •  +  $  y*-s  +  !  y-«  +  $  y»-i  +  f  y*]. 
Adding, 

2  A  =h[ty0+Wyi+2y2  +  2y3  +  .  .  .  +2yn_3  +  2yn_2  +^yn 
*  Given  by  Prof.  Durand  in  Engineering  News,  Jan.,  1894. 


ART.  102 


APPLICATIONS  OF  APPROXIMATE   RULES 


227 


Hence, 

AD  =  h  [A  (yo  +  y»)  +  H  (yi  +  y-i)  -f  » 
=  A  [0.4  (jo  +  yw)  +  i.i  (yi  +  y*-i)  +  ^ 
Collecting  our  rules,  we  have 

(1)  AR  =  h(y0  +  yi  +  yz+  •  -  -  +  yn-i), 
or          AR'  =  h(yi  +  y2  +  y3+  •  •  •  +  yn). 

(2)  4r  =  A  ft  On,  +  yn)  +  yi  +  y2  +  •  •  • 

(3)  -4s  =  I  ^  [(yo  +  yn)  +  4  (yi  +  y»  +  ys  + 

+  2  (ys  + 

(4)  AD  =  h  [0.4  Cvo  +  y»)  +  i.i  (yi  +  yn- 

102.   Applications  of  approximate  rules.  —  We  shall  give  some  ex- 
amples illustrating  the  application  of  these  rules. 

Cl°dx 
I.   Area.  —  Evaluate   I     —  .     This  is  equivalent  to  finding  the  area 

*J  2        X 

between  the  curve  y  =  i/x,  the  x-axis,  and  the  ordinates  x  =  2  and 
x  =  10.  If  we  divide  the  interval  into  8  parts,  then  h  =  I  ;  we  have  the 
table 


•  •  +  y*- 

+  ye  +  • 


+  ?»-»)]• 


X 

2               3 

4 

5 

o 

7 

8           9 

10 

y 

i 

4 

i 

i 

I 

1           i 

iV 

AR  = 

i  (i  +  1  + 

i  +  • 

•  +*} 

=  1.8290; 

AR  = 

i  (i  +  1  + 

i+  • 

•  +  iV)  =  14290; 

AT  = 

i  B  (H-  A)  +  *+••• 

+  *]- 

1.6290; 

i.i 


4^  =  i  [0.4  (i  +  TV) 

ri0  dx 

By  actual  integration,    /      —  = 

t/2        X 


In  x 


*)+2(t  +  i  +  «]=»  1.6109; 

i  +  i  +  •  •  •  +  1]  =  1.6134- 

=  In  io  —  In  2  =  In  5  =  1.6094. 


We  note  that  Simpson's  rule  gives  the  best  approximation  (within 
o.i  %  of  the  true  value),  with  Durand's  next. 
If  we  take  h  =  \, 


Thus  the  Trapezoidal  rule  with  16  ordinates  does  not  give  the  accuracy 
given  by  Simpson's  rule  with  8  ordinates. 

2.   Area.  —  The  half-ordinates  in  feet  of  the  mid-ship  section  of  a 
vessel  are 

12.5,  12.8,  12.9,  13.0,  13.0,  12.8,  12.4,  ii.  8,  10.4,  6.8,  0.5, 
and  the  ordinates  are  2  feet  apart;  find  the  area  of  the  whole  section. 


228       APPROXIMATE  INTEGRATION  AND   DIFFERENTIATION        CHAP.  IX 


\AT  =  2  ft  (12.5  +  0.5)  +  12.8  +  .  .  .  +  6.8]  =  224.8; 
±A8  =  l  [(12.5  +  0.5)  +  4  (12.8  +  13.0  +  12.8  +  1 1.8  +  6.8) 

+  2  (12.9  +  13.0  +  12.4  +  10.4)] 

Hence,  AT  =  449.6  sq.  ft.,     As  =  452.2  sq.  ft. 

3.    Work.  —  Given  the  following  data  for  steam 


226.3 


V 

2 

4 

6 

8 

10 

p 

68.7 

3i-3 

19-8           14-3 

n-3 

where  v  is  the  volume  in  cu.  ft.  per  pound  and  p  is  the  pressure  in  pounds 
per  sq.  in.;  find  the  work  done  by  the  piston. 

Work  =    /     p  dv  ;  this  is  equivalent  to  finding  the  area  under  the  curve 

obtained  by  plotting  (v,  p). 

WT  =  2  ft  (68.7  +  11.3)  +  31.3  +  19.8  +  14.3]  =  210.80; 
Ws  =  .f  [(68.7  +  11.3)  +  4  (31.3  +  14.3)  +  2  (19.8)]  =  201.33. 

By  the  methods  of  Chapter  VI  we  find  the  empirical  formula  con- 
necting v  and  p  to  be  pv1-12  =  148,  and  hence, 


W 


rw  p 

=l     pdv  =  i4SI 

t/2  Jz 


=  148 


.,-0.12 
- 

—  O.I2 


110 

\2 


199.31 


This  last  value  differs  from  the  value  given  by  Simpson's  rule  by  about  I  %. 

4.    Mean  effective  pressure.     Indicator  diagram.     Fig.   1020  is  a  re- 

production of  an  indicator  diagram;  to  find  the  mean  effective  pressure. 


FIG.  1020. 

The  mean  effective  pressure  P  is  the  area  of  the  diagram  divided  by 
the  length  of  the  diagram,  since  the  area  represents  the  effective  area  of 
the  piston  in  sq.  in.  and  the  length  represents  the  length  of  the  stroke  in  ft. 
Since  the  total  area  enclosed  by  the  curve  is  the  difference  between  the 
area  bounded  by  a  horizontal  axis,  the  end  ordinates,  and  the  upper  part 
of  the  curve,  and  the  area  bounded  by  the  same  straight  lines  and  the 
lower  part  of  the  curve,  we  need  merely  measure  the  lengths  of  the  ordi- 
nates within  the  curve.  The  diagram  is  3.5  ins.  long.  We  divide  the 
interval  into  8  parts;  then  h  =  T7ff,  and  we  measure  the  ordinates 

o,       0.40,       0.63,       0.91,       0.98,       i.  oo,       0.92,       0.74,       o. 


ART   102 


APPLICATIONS  OF  APPROXIMATE   RULES 


229 


yV  [040  +  0.63  +  •  •  •   +  0.74]  =  2.44; 

?7s  [4  (0.40  +  0.91  +  1. 00  +  0.74)  +  2  (0.63  +  0.98  +  0.92)] 


2.52. 


p  —      -s 

3-5 


3-5 


J,  and  we  measure  the 


We  divide  the  interval  into  14  parts;  then  h 
ordinates 

o,  0.30,  0.42,  0.54,  0.68,  0.88,  0.96,  0.98,  i.oo,  i. 02,  0.97,  0.89,  0.78,  0.64,  o. 
AT  =  I  [0.30  +  0.42  +  •   •  •  +  0.64]  =  2.52. 
As  =  &  [4(0-30+0.54 H +o.64)+2  (0.42+0.68+  •  •  •  +0.78)1  =  2.55. 

Hence,  /  /  P  =  ff  ->-&  -  0.73. 

We  note  that  As  with  9  ordinates  has  the  same  value  as  AT  with  15 
ordinates. 

5.  Velocity.  —  Given  a  weight  of  1000 
tons  sliding  down  a  i%  grade  (Fig.  1026) 
with  a  frictional  resistance  of  10  Ibs.  per  ton 
at  all  speeds.  The  total  resistance  is  30,000 
Ibs.  (a  frictional  resistance  of  10,000  Ibs.  and 
a  grade  resistance  of  20,000  Ibs.).  Let  the  following  table  express  the 
accelerated  force  F  as  a  function  of  the  time  /  in  seconds : 


joo> 
FIG.  IO2&. 


100       200      300  1 400 1    500   I   600   I    700 


800 


900 


F  |2o,ooo|  19,000  16,000  ii,ooo|5oooj  — loooj— 50001—8500  — 

Find  the  velocity  acquired  by  the  body  in  1000  seconds. 

2,000,000      1,000,000 


Since       F  =  m  X  a,     and     m 


therefore,      a  =  —  = 
m 


16.1  F 


g 

.     dv 
and     dt 


1,000,000 

We  form  a  table  for  the  acceleration  a. 


a, 


(0.322 


0.306 


200        300 


0.258  |  0.177 


400    I       500 


O.oSl     —O.OI6 


600 


700 


16.1 
hence,    v 

800    ! 


-0.081 1  —0.137 


-£ 


900 


IOOO 

-I5~ooo 


adt. 


—0.177 1  -0.209 1-0.242 


Here,  h  =  100,  so  that 

VT=  ioo  [£(0.322—  0.242)  +  (0.306+0.258+  •  •  •  —0.209)]  =24.2  ft.  per  sec. 

*>s  =  A§-  [(0.322  —  0.242)  +  4  (0.306  +  0.177  —  0.016  —  0.137  —  0.209) 

+  2  (0.258  +  0.081  —  0.081  —  0.177)]  =  24.2  ft.  per  sec. 

6.    Volume.  —  If  Sx  is  the  area  of  a  cross-section  of  a  solid  made  by  a 

plane  perpendicular  to  the  .T-axis,  then  the  volume  of  the  solid  included 

between  the  planes  #0  and  xn  is  V  =    I  " Sxdx.     In  order  to  integrate, 

Jx, 

we  must  know  the  analytical  expression  for  Sz  as  a  function  of  x. 
Otherwise  we  employ  the  approximate  formulas;  the  values  of  Sx  are  the 
ordinates  and  h  is  the  common  distance  between  the  cutting  planes. 


230       APPROXIMATE   INTEGRATION   AND    DIFFERENTIATION         CHAP.  IX 


A  buoy  is  in  the  form  of  a  solid  of  revolution  with  its  axis  vertical, 
and  D  is  the  diameter  in  ft.  at  a  depth  p  ft.  below  the  surface  of  the  water. 


P 

o 

0-3 

0.6 

0.9 

1.2 

i-5 

1.8 

D 

6.00 

5-90 

5.80 

5-55 

5-25 

4.70 

4.20 

D2 

36.00 

34.81 

33-64 

30.80 

27.56 

22.09 

17.64 

Find  the  weight  of  water  displaced  by  the  buoy  (i  cu.  ft.  of  sea  water 
weighs  64.11  Ibs.). 


Here, 


X1.8- 
-D*dp,     and     h  =  0.3, 
4 


0.3  7T 


hence,      Vs=  —J--  [(36.00  +  17.64)  +  4  (34.81  +  30.80  +  22.09) 

O      T" 

+  2  (33.64  +  27.56)]  =  41.38  CU.  ft., 

and  the  weight  of  water  displaced  =  2652.87  Ibs. 

The  areas  in  sq.  ft.  of  the  sections  of  a  ship  below  the  load-water  plane 
and  3  ft.  apart  are 

7500,         7150,         6640,         5680,        4225,         2430,         260, 

where  the  load-water  plane  has  an  area  of  7500  sq.  ft.     Find  the  dis- 
placement in  tons  (35  cu.  ft.  of  sea  water  weigh  I  ton). 

VT'-=3  [K7500+260)  +  (7150+6640+5680+4225+2430)]  =  90,015  cu.  ft. 
Vs  =  f  [(75<>°+26o)  +4(7150+5680+2430)  +2(6640+4225)]  =  90,53001.  ft. 

Hence,  the  displacement  is  2572  tons  by  the  Trapezoidal  rule  and  2587 
tons  by  Simpson's  rule. 

7.    Moment  of  inertia.  —  The  moments  of  inertia  of  an  area  about  the 
axes  are 

/*=      **l?dx,        /, 


The  evaluation  of  these  integrals  is  equivalent  to  finding  the  areas  under 
the  curves  with  £  y3  or  x2y  as  ordinates  and  x  as  abscissas. 

The  half-ordinates  in  ft.  of  the  mid-ship  section  of  a  vessel  are 

12.5,    12.8,     12.9,    13.0,    13.0,    12.8,     12.4,     ii.  8,     10.4,    6.8,    0.5, 

and  the  ordinates  are  2  ft.  apart.     Find  the  moment  of  inertia  of  the 
entire  section  about  the  axis. 

Here,    Jx  =  2    /     ^  y3  dx,     h  =  2,     and  the  values  of  r3  are 
Jo 

1953.1,  20972,  2146.7,  2197.0,  2197.0,  2097.2,   1906.6,   1643.0,  1124.9,  314.4,  o.i, 

and  applying  Simpson's  rule, 

J*  =  !  (I)  [(I953-I  +  o.i)  +  4  (2097.2  +  •  •  •  +  314.4) 

+  2  (2146.7  +    •    •    .    +  1124.9)]  =    22,266.1. 


ART.  103       GENERAL  FORMULA  FOR  APPROXIMATE   INTEGRATION       231 

8.   Pressure  and  center  of  pressure.  —  The  pressure  on  a  plane  area 
perpendicular  to  the  surface  of  the  liquid,  between  depths  x0  and  xn,  is 

p  =  w   I  n  xydx,  where  w  is  the  weight  of  the  liquid  per  unit  volume,  y  is 

t/*c 

the  width  of  the  area  at  a  depth  x  beneath  the  surface.  The  depth  of 


the  center  of  pressure  of  such  an  area  is  given  by  x 


/•xn 

I     xyd 
«/*o         ! 


All 


these  integrals  can  be  evaluated  approximately. 

9.    Center  of  gravity.  —  The  coordinates  of  the  center  of  gravity  of  an 
area  are 


/* 


Moment  about  OY 
Area 


f 

=J- 


T 

fydx 


Moment  about  OX 
Area 


The  half-ordinates  in  ft.  of  the  mid-ship  section  of  a  vessel  are       '< 
12.5,    12.8,    12.9,    13.0,    13.0,    12.8,    12.4,    ii. 8,     10.4,    6.8,    0.5, 

and  the  ordinates  are  2  ft.  apart.     Find  the  center  of  gravity  of  the 
section. 


xydx 


Moment  about  OY 


/' 

/ 

Jo 


ydx 


Area 


and  applying  Simpson's  rule  to  the  table, 


x 

0 

2 

4 

y 

12.5 

12.8 

12.9 

xy 

0 

25.6 

51.6 

I3-Q 
78.0 

Ms  =  §  [(o  +10.0)  +4  (25.6  + 

=  2018.9. 
As  =  I  [(12.5 +  0.5) +4  (12.8+  •  •  • 

=  226.1. 

_      2018.9 
Hence,  x  =  — 


8 

10 

12 

14 

16 

18 

20 

13.0 

12.8 

12-4 

1  1.  8 

10.4 

6.8 

o-5 

104.0 

128.0 

148.8 

165.2 

166.4 

122.4 

IO.O 

122.4) +2  (51.6+  • 
+  6.8) +2  (12.9  + 

8-93  ft. 


+  166.4)] 
•  +10.4)] 


103.  General  formula  for  approximate  integration.  —  We  may  derive 
a  general  formula  for  approximate  integration  by  integrating  any  of  the 
formulas  of  interpolation.  Thus,  Newton's  formula  (p.  215), 


naQ 


n  (n  —  i) 

' 


n  (n  -  i)  .  .  .  (n  -  k  +  i) 

~~  °' 


232       APPROXIMATE   INTEGRATION  AND   DIFFERENTIATION        CHAP.  IX 

where  x  =  XQ  +  nh,  is  true  for  all  values  of  n  if  some  order  of  differences 
is  constant  or  approximately  constant.  Multiplying  by  dn  and  inte- 
grating term  by  term  between  the  limits  o  and  n,  we  have 

r*  "*'          Cn  Cn         b0  cn 

I    yndn  =  y0  I    dn  +  a0  I     n  dn  +  -  /     n  (n  -  i)  dr 

»/0  t/O  vO  [?  *SO 


Since  x  =  XQ  +  nh,  therefore,  n  =  —  ;  —  °     and    dn  =  -,-  dx.     Hence, 

n  n 


Thus,  if  the  differences  after  some  order,  as  the  &th,  are  negligible,  we 
may  use  this  formula  to  get  the  approximate  area  between  the  curve,  the 
x-axis,  and  the  ordinates  x  =  x0  and  x  =  xn.  The  process  is  equivalent  to 
approximating  the  equation  of  the  curve  by  a  polynomial  of  the  &th 
degree.  The  differences  a0,  b0,  c0,  .  .  .  are  those  which  occur  in  a  line 
through  y0  parallel  to  the  upper  side  of  the  triangle  in  the  scheme  on  p.  210. 
Similar  integration  formulas  can  be  derived  from  the  other  interpolation 
formulas. 

If  the  interval  from  x0  to  xn  is  large,  it  is  well  to  divide  this  into  smaller 
intervals,  apply  the  formula  to  each  of  the  smaller  intervals,  and  add  the 
results.  In  this  way  we  may  derive  the  formulas  of  Art.  101  and  similar 
formulas  as  special  cases  of  the  above  general  formula. 

Let  us  first  note  that  by  means  of  the  rule  for  the  formation  of  the 
successive  differences  of  a  function  (p.  210)  we  may  express  the  differences 
do,  b0,  CQ,  .  .  .  in  terms  of  y0,  yi,  yz,  -  -  •  •  Thus, 

ao  =  y\  —  yo, 

bo  =  ai  —  a0  =  (j2  —  yi)  —  (yi  —  yo)  =  yz  —  2  yi  +  yo, 

c0  =  bi-b0  =  [(y3  -  2yz  +  y1)  -  (y2'-2yi 

do  =  y±  —  4  y3  +  6  yt  —  4  yi  +  y0, 

e*  =  Js  -  5  ^4  +  10  ;y3  -  10  y2  +  5  yi  -  y0, 


,  - 

ko  =  yk  -  kyk-i  H  --  T-  -  yk-z  -   •  •  • 

where  the  coefficients  in  the  right  members  of  these  equations  are  the 
binomial  coefficients,  taken  alternately  plus  and  minus. 


ART.  103         GENERAL  FORMULA   FOR  APPROXIMATE   INTEGRATION      233 

(i)  Let  n  —  I  and  b0,  c0,  .  .  .  all  zero,  i.e.,  approximate  the  curve 
(Fig.  ioia)  from  x0  to  Xi  by  a  straight  line,  y  =  A  +  Bx.    Then 


f 
«y* 


Xlydx  =  A  [yo  +  iflo]  =  *  bo 

J 


Applying  this  result  to  each  interval  and  adding,  we  get  the  Trapezoidal 
rule: 

AT 

(2)  Let  n  =  2  and  c0,  J0,  •  •  •  all  zero,  *.e.,  approximate  the  curve 
(Fig.  ioia)  from  x0  to  xz  by  a  parabola,  y  =  A  +  -Brc  +  Cx2.     Then 


/\T 

I 


y  <fcc  =  A  [2  y0  +  2  Co  +  $  fto]  =  A  [2  y0  +  2  (yi  -  y0)  +  $  (y2  -  2  yi  +  y0)] 

A 
=  -  bo  +  4  yi  +  y»]- 

O  , 

Applying  this  result  to  an  even  number  of  intervals,  two  at  a  time,  and 
adding,  we  get  Simpson's  rule: 

As= 

(3)  Let  n  =  3  and  d0,  c0,  .  .  .  all  zero,  i.e.,  approximate  the  curve 
(Fig.  ioia)  from  x0  to  #3  by  a  parabola  of  the  3d  degree,  y  =  A  +  Bx  + 
Cjc2  +  Dx3.  Then 

£'y  dx  =  A  [3  y0+l  ao+l  &o+!  Co]  =  A  [3  yo+l  (yi-yo)  +  I  (^2- 
. 

+  I  (ys  -  3  ?2  +  3  yi  -  3>o)]  =  t  ^  bo  +  3  yi  + 

Applying  this  result  to  n  intervals,  where  n  is  a  multiple  of  3,  and  adding, 
we  get  Simpson's  three-eighths  rule: 

As'= 


(4)  Let  n  =  6  and  the  differences  beyond  the  6th  order  negligible, 
i.e.,  approximate  the  curve  (Fig.  ioia)  from  x0  to  x6  by  a  parabola  of  the 
6th  degree,  y  =  A  +  Bx  +  Cx2  +  •  •  •  +  Hx6.  Then 


r 


ydx  =  h[6y0+  i8a0  +  27  60 


Substituting  the  values  of  Oo,  &o»  •  •  •  >  /o  in  terms  of  the  y's  and  re- 
placing T*iV  /o  by  T^J  /0,  thus  neglecting  fiofo  which  will  be  fairly  small,  we 
get  Weddle's  rule: 

Aw  =    I   'ydx  =  fV  h  [yo  +  5 yi  +  ^2  +  6y3  +  y4  +  5  J?>  +  ye]. 

Jxo 

We  may  apply  this  rule  to  n  intervals  where  n  is  a  multiple  of  6. 


234       APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION        CHAP.  IX 

r6dx 
— . 
x 

We  divide  the  interval  into  6  equal  parts,  so  that  h  =  o.i.     From  the 
table 


2 

2.1 

2.2 

2-3 

2.4 

2.5 

2.6 

I 

2 

I 
2.1 

I 
2.2 

I 

2^3 

I 
2.4 

I 
2^5 

I 

2^6 

=  o.i  \-  (-  +  -^  +  —  +  —  +  —  +  —  +  —  1  =  0.2624493; 

|_2  \2        2.6/        2.1        2.2        2-3  ^  2.4        2.5] 


+          =  0.2623645; 


A,  -     (o.l)+5.+.+6.++5+..   =  0.2623643. 


By  integration, 


r*—= 

J2      x 


\nx\     =ln2.6  — In2  =  ln  1.3=0.2623637. 


AT  agrees  with  A  to  4  decimals,  while  AS,  AS',  and  ^4^  agree  about 
equally  well  with  A  to  6  decimals. 

104.   Numerical  differentiation.  —  We  are  to  find  the  slope  of  the 
curve  y  =  f(x)  at  any  point  when  the  curve  is  drawn  or  a  table  of  values 

of  equidistant  ordinates  are  given,  i.e.,  we  are  to  find  ~  when  the  analyti- 
cal form  of  the  function  is  unknown.  Graphically,  we  must  construct 
the  tangent  line  to  the  curve  at  the  given  point.  The  exact  or  even 
approximate  construction  of  the  tangent  line  to  a  curve  (except  for  the 
parabola)  is  difficult  and  inaccurate.* 

dy 

We  may  derive  an  expression  for  -j-  by  differentiating  Newton's  in- 
terpolation formula.     Newton's  formula 


. 

yn  =  yo  -f-  na0  H 


n  (n  —  i) 


n  (n  -  i)  .  .  .  (n  -  k  +  i) 
\k 


k0, 


is  true  for  all  values  of  n  if  some  order  of  differences,  as  the  &th,  is  con- 
stant or  approximately  constant. 

Since  x  =  XQ  +  nh,  therefore,  dx  —  hdn,  and  —-  =  T  ~r-  .  -=^  =  —  —  ^. 

2        *      * 


—-      T  ~r-  . 
dx      h  dn    dx2 

*  See  Art.  106  on  graphical  differentiation. 


— 
dn* 


ART.  104  NUMERICAL  DIFFERENTIATION  235 

Hence, 


The  values  of  these  coefficients  are  tabulated  for  values  of  n  between 
O  and  I  at  intervals  of  o.oi.* 

For  the  tabulated  values  x0,  Xi,  .  .  .  ,  xn,  we  have  n  =  o,  so  that  for 
these  values  of  x  we  have  the  simpler  formulas 

dy 


If  the  value  of  x  for  which  -7-  is  required  is  near  the  end  of  the  table, 

we  may  use  similar  formulas  derived  from  the  modified  Newton's  formula 
for  end-interpolation  (p.  217). 

d'V          d?'V 
Example.      Find  -j-  and  -7-^  for  x  =  3  and  x  =  3.3  from  table  (l)  on 

p.  211  and  check  the  results  by  differentiating  y  =  x3. 

Since  x  =  3  is  a  tabulated  value  we  apply  the  second  set  of  formulas  : 

|  =  [37-2'(24)+f(6)]=27;      g-  [24-  6]  =  18. 
From  y  =  x3,     ^  =  3^  =  27,     £|  =  6x  =  18. 

For  x  =  3.3  we  apply  the  first  set  of  formulas,  where  #0  =  37.  &o  =  24, 
c0  =  6,  n  =  0.3.  Then 

^  =  [37  +  (-  04)^+  (0-47)  f]=  32.67;  g  =  [24  +  (-0.7)  6]=  19.8. 
From  y  =  x*,     ^  =  3  x*  =  32.67,     g  =  6  x  =  19.8. 

Example.  Rate  of  change.  —  The  following  table  gives  the  results  of  ob- 
servation ;  6  is  the  observed  temperature  in  degrees  Centigrade  of  a  vessel 
of  cooling  water,  t  is  the  time  in  minutes  from  the  beginning  of  observation. 


92.0 


85-3 


79-5     I     74-5    I     70-2 


To  find  the  approximate  rate  of  cooling  when  t  =  I  and  /  =  2.5. 
*  See  Rice,  Theory  and  Practice  of  Interpolation. 


236       APPROXIMATE  INTEGRATION  AND   DIFFERENTIATION        CHAP.  IX 
From  the  table  of  differences 


/ 

e 

Ai 

A1 

A> 

o 

92.0 

I 

85-3 

-6.7 

0.9 

-5-8 

—  O.I 

2 

79-5 

0.8 

-5-0 

—  O.I 

3 

74-5 

0.7 

-4-3 

0.4 

4 

70.2 

I.I 

-3-2 

5 

67.0 

when/=i,  w  =  o    and          =     -5.8  -^  (0.8)  +^  (-o.i)    =  -6.23; 
when  /  =  2.5,  n  =  0.5  and^  =  [-5.0  +  o  +  (-  0.25)  f^4)]  — S-oa. 

Example.  Maximum  and  minimum.  —  The  following  table  gives  the 
results  of  measurements  made  on  a  magnetization  curve  of  iron;  B  is  the 
number  of  kilolines  per  sq.  cm.,  n  is  the  permeability  (Fig.  104). 


o 

I 

2 

3 

4 

5 

6 

7 

8    I    9 

IO    1    II    1    12 

1120 

14       i« 

370 
too 

WO 
100 

wo 

/*) 

>00 

too 
wo 

°( 

57C 

>   73< 

)   865 

985 

1090 

H75 

1245 

1295!  1330 

I34o|i32o|i25o 

930     72 

^ 

••v 

^ 

X 

x^ 

X 

\ 

S 

/ 

\ 

/ 

/ 

\ 

\ 

/ 

)      1       2      3 

4 

5 

6 

7 

FIG. 

8      9 
(B) 

104. 

10     11     12 

13     If     15 

To  find  the  maximum  permeability.  In  Fig.  104  the  maximum  perme- 
ability appears  to  be  in  the  neighborhood  of  B  =  10.  We  therefore  tabu- 
late the  differences  of  /*  in  the  neighborhood  of  B  =  10. 


1330 
1340 
1320 
1250 
1 1 2O 


-  20   " 

-  70   _ 
-130 


ART.  105  -    GRAPHICAL   INTEGRATION 

For  values  of  B  between  B  =  9  and  B  =  10,  we  have 


237 


For  a  maximum,    TB  =  °>    hence   6n?  +  6n  —  n  =  o,    and    n  =  0.94. 

Therefore,  B  =  B0  +  nh  =  9.94. 

We  find  the  corresponding  value  of  p  by  the  interpolation  formula, 
M  =  1330  +  (0.94)  (10)  +  (0.0282)  (-30)  +  (o.oioo)  (-20)  =   1340. 
If  we  take  account  of  A1  and  A2  only,  we  get 

—  —  J  =o,     or     n  =  |  =  0.83,     and     B  =  9.83. 

Then          /*  =  1336  +  (0.83)  (10)  +  (0.0275)  (-30)  =  1337-5- 

\ 
105.   Graphical  integration.  —  Let  us  find  the  value  of  the  definite 

integral     /   f(x)  dx  or  the  area  under  the  curve  y  =  f(x)  by  graphical 

methods.     We  draw  the  curve  y  =  f(x)  (Fig.  1050)  and  along  the  ordinate 
at  P  (x,  y}  erect  the  ordinate  y'  whose  value  is  a  measure  of  the  area  under 


FIG.  1050. 


Fio.  1056. 


the  curve  y  =  f(x)  from  the  initial  point  A  (x  =  a)  to  the  point  P,  i.e., 
y'  =  I  }(x)  dx.  Thus  for  every  point  P  (x,  y)  we  have  a  corresponding 

point  P'  (x,  y').  The  curve  traced  by  the  point  P'  (marked  /  in  the 
figure)  is  called  the  integral  curve  and  the  curve  traced  by  the  point  P 
(marked  A  in  the  figure)  is  called  the  derivative  curve.  Evidently,  if  P 
and  Q  are  two  points  on  the  A-curve  and  P'  and  Q'  are  their  correspond- 
ing points  on  the  /-curve,  the  difference  of  the  ordinates  of  P'  and  Q', 
y"  —  y',  is  a  measure  of  the  area  under  the  arc  PQ. 

The  practical  construction  of  the  integral  curve  consists  of  the  follow- 
ing steps  (Fig.  1056). 

(i)  Divide  the  interval  from  XQ  to  xn  into  n  equal  or  unequal  intervals 
and  erect  the  ordinates  y0l  y\,  .  .  .  ,  yn. 


238       APPROXIMATE  INTEGRATION  AND   DIFFERENTIATION        CHAP.  IX 


(2)  Measure  the  areas  xoA0AiXi  =  yi,  x0A oAzx2  =  yz',  •  •  .  , 
XoA0Anxn  =  yn'.  These  areas  may  be  found  by  means  of  a  planimeter 
or  by  the  construction  of  the  mean  ordinates.  Thus,  the  area  x0A0AiXi 
is  equal  to  the  area  of  a  rectangle  whose  base  is  x0Xi  and  whose  altitude 
is  the  mean  ordinate  mi  within  that  area.  Similarly,  the  area  XiAiA&i 
is  equal  to  the  area  of  a  rectangle  whose  base  is  x\x2  and  whose  altitude  is 
the  mean  ordinate  m2  within  that  area.  Estimate  the  mean  ordinates 
mi,  mz,  m3,  .  .  .  ,  mn  within  the  successive  sections.  Then 
y\  =  mi  (*o*i),  yz  =  y\  +  m?  (xix2),  y*  =  y2f  +  m$  (xyxa),  .  .  .  , 

ynf  =  yn'-i  +  mn  (*n-i*n). 

If  the  intervals  are  all  equal,  i.e.,  x0Xi  =  XiX2  =  .  .  .  =  xn-\xn  =  AJC, 
then  y'  =  2m AJC.  (We  shall  later  give  a  more  exact  construction  for  the 
mean  ordinate.) 

(3)  At  xi,  x2>  x3,  .  .  .  ,  xn  erect  or- 
dinates xiBi,  XzBz,  .  .  .  ,  XnBn  equal 
respectively  to  y\  ,  y2f,  .  .  .  ,  yn',  and 
draw  a  smooth  curve  through  the  points 
Bo,  Bi,  B2,  .  .  .  ,  Bn.  This  last  curve 
will  approximate  the  required  integral 
curve. 

Example.  Construct  the  integral 
curve  of  the  straight  line  y  =  I  —  x  be- 
tween x  =  o  and  x  =  2.  (Fig.  105^.) 

Divide  the  interval  from  x  =  o  to  x  =  2 
into  10  equal  parts  and  erect  the  ordinates 
given  in  the  table;  here,  AJC  =  0.2. 


FIG. 


* 

y 

« 

mAx 

/  =  2mA* 

0.9 

0.18 

o 
o  18 

.6 

.8 

.0 

.6 

.4 

.2 
—      .2 

—    -4 
-    .6 
-    .8 

—     .0 

0.7 
0.5 
0-3 

O.I 
—  O.I 

-0.3 
-0.5 
-0.7 
-0.9 

o.  14 

O  .  IO 

0.06 

0.02 

—  o  .02 
—  o  .06 

—  O.  IO 

—0.14 
-0.18 

0.32 
0.42 
0.48 
0.50 
0.48 
0.42 

0.32 

0.18 

0 

It  is  evident  that  the  mean  ordinate  in  each  section  is  merely  one-half 
the  sum  of  the  end  ordinates,  so  that  the  values  of  m  are  easily  found. 
Erect  the  ordinates  y'  and  draw  a  smooth  curve  through  the  ends  of  the 

/'z 

ordinates.     The  curve  will  approximate  the  parabola  y'  =  I     (i  —  x)  dx 

*/o  _ 


ART.  105 


GRAPHICAL  INTEGRATION 


239 


Example.  The  following  table  gives  the  accelerations  a  of  a  body 
sliding  down  an  inclined  plane  at  various  times  /,  in  seconds.  To  find 
the  velocity  and  distance  traversed  at  any  time,  if  the  initial  velocity 
and  initial  distance  are  zero. 


o    I  IPO 
0.320!  0.304 


200  I   300   I  400 


0.256)  0.176  |  0.080 


500 


— 0.016 


600 


—0.080 


700     I     800 
—0.136!  —0.176 


900 


-0.208 


1000 


—0.240 


Since  v  =    I  a  di  and  s  =    /  v  dt,  the  time- velocity  curve  is  the  integral 

curve  of  the  time-acceleration  curve,  and  the  time-distance  curve  is  in 
turn  the  integral  curve  of  the  time-velocity  curve. 

In  Fig.  105^,  we  have  plotted  t  as  abscissas  and. a  as  ordinates.     Th.- 
units  chosen  are  I  in.  =  100  sec.,  and  I  in.  =  0.16  ft.  per  sec.  per  sec. 


0.32 
0.16 

0 

(a) 

-0.16 
-0.32 

JU.UUV 
60  000 

^ 

^ 

' 

y 

X 

50,000 
40,000 
30,000 
20,000 
10,000 
n 

V 

t 

^ 

00 

—  --», 

\ 

^ 

^  —  • 

—  ^, 

-^ 

/ 

^^ 

X 

S 

"^ 

x 

80 

60 

/ 

N 

X 

/ 

^ 

s 

/ 

X 

v^ 

/ 

^^ 

\ 

• 

/ 

/ 

/ 

xl 

^ 

^ 

\ 

A 

? 

••. 

/ 

^ 

^^ 

\ 

\ 

40 

* 

/ 

^ 

"  —  -^ 

1  —  —» 

1  —  -». 

\ 

/ 

^ 

? 

20 

/ 

X 

/ 

-• 

" 

' 

0 

JOO       200      300      400      500       600       700      800       900     1000 
FIG.  I05</. 


/ 

a 

avg.  ace.  am 

am&t 

v  =  2flmA< 

avg.  vel.  f.m 

vm&t 

5  =  2vm&l 

o 

IOO 
200 
300 
400 
500 

6oc 
700 
800 
900 

IOOO 

0.320 
0.304 
0.256 
0.176 
0.080 
—  0.016 
-0.080 
—0.136 
—0.176 
—0.208 
—0.240 

0.312 

0.280 
0.216 
0.128 
0.032 
—  0.048 
—  0.108 
—  0.156 
—  0.192 
—0.224 

31-2 
28.0 

21.6 
12.8 
3-2 

-  4.8 
-10.8 
-15.6 
-19.2 
—  22.4 

0 

31-2 

59-2 
80.8 
93-6 
96.8 
92.0 
81.2 
65.6 
46-4 
24.0 

15-6 
45-2 
70.0 
87.2 
9S-2 
94-4 
86.6 
73-4 
56.0 
35-2 

1560 

4520 

7000 
8720 
9520 

9440 

8660 

7340 
5600 

3520 

0 

1,560 
6,080 
13.080 
21,800 
31.320 
40,760 

49.420 
56,760 
62,360 
65,880 

240       APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION        CHAP.  IX 

In  each  interval  of  100  sec.  we  have  estimated  the  mean  acceleration 
as  the  average  of  the  accelerations 'at  the  beginning  and  end  of  the  in- 
terval; thus,  in  the  first  interval,  am  =  |  (0.320  +  0.304)  =  0.312.  This 
is  equivalent  to  replacing  the  arcs  of  the  curve  by  their  chords  or  to  find- 
ing the  area  by  the  trapezoidal  rule.  Since  the  initial  velocity  is  zero, 
the  (t,  v)  curve  joins  /  =  o,  v  =  o  with  /  =  100,  v  =  31.2,  etc.  We  have 
drawn  the  (t,  v)  curve  with  a  unit  of  I  in.  =  20  ft.  sec. 

In  each  interval  of  100  sec.  we  have  estimated  the  mean  velocity  as  the 
average  of  the  velocities  at  the  beginning  and  end  of  the  interval;  thus  in 
the  first  interval,  vm  =  \  (o  +  31.2)  =  15.6.  Since  the  initial  distance 
is  zero,  the  (/,  s)  curve  is  drawn  through  the  points  t  =  o,  s  =  o,  t  =  100, 
s  =  1560,  etc.  The  unit  chosen  is  I  in.  =  10,000  ft. 

The  tables  for  v  and  5  give  the  velocity  and  distance  at  the  end  of  each 
100  seconds,  and  we  may  interpolate  graphically  or  numerically  for  the 
velocity  and  distance  at  any  time  between  t  =  o  and  /  =  1000. 

In  the  foregoing  discussion  the  accuracy  of  the  construction  of  the 
integral  curve  depends  largely  upon  the  construction  of  the  mean  ordi- 
nates  in  the  successive  intervals.  If  the  intervals  are  very  small,  we  may 
get  ftie  required  degree  of  accuracy  by  replacing 
the  arcs  by  their  chords  and  taking  for  the  mean 
ordinate  the  average  of  the  end  ordinates. 

The  approximation  of  the  mean  ordinate 
for  the  arc  AQA\  (Fig.  1050)  is  equivalent  to 
finding  a  point  M  on  the  arc  such  that  the  area 
under  the  horizontal  CoCi  through  M  is  equal 
to  the  area  under  the  arc  A0Ai  or  such  that  the 
shaded  areas  A0C0M  and  A\C\M  are  equal.  By  means  of  a  strip  of  cellu- 
loid and  with  a  little  practice  the  eye  will  find  the  position  of  M  quite 
accurately,  for  the  eye  is  very  sensitive  to  differences  in  small  areas. 

We  may  draw  the  integral  curve  by  a  purely  graphical  process.  Let 
us  first  consider  the  case  when  the  derivative  curve  is  the  straight  line 
AB  parallel  to  the  X-SLXIS  (Fig.  IO5/).  Choose  a  fixed  point  5  at  any  con- 
venient distance  a  to  the  left  of  0.  Extend  AB  to  the  point  K  on  the  y- 
axis  and  draw  SK.  Through  A'  (the  projection  of  A  on  the  re-axis)  draw 
a  line  parallel  to  SK  cutting  the  vertical  through  B  in  B' '.  Then,  the 
oblique  line  A'B'  is  the  integral  curve  of  the  horizontal  line  AB.  For,  if 
P  and  P'  are  two  corresponding  points,  then 

y'  :  A'Q  -  y0  :  a,  or  /  =  ^  (y0  X  A'Q)  =  ^  X  (area  under  AP). 

Similarly,  for  another  horizontal  CD,  with  C  and  B  in  the  same  verti- 
cal line,  extend  CD  to  the  point  L  on  the  y-axis  and  draw  SL;  through  B" 
draw  a  line  parallel  to  SL  cutting  the  vertical  through  D  in  C"\  then,  the 
oblique  line  B"C"  is  the  integral  curve  of  the  horizontal  CD.  Finally, 


ART.  105 


GRAPHICAL  INTEGRATION 


24I 


draw  B'C'  parallel  to  B"C"  or  to  SL\  then  the  broken  oblique  line  A'B'C 
is  the  integral  curve  of  the  broken  horizontal  line  ABCD. 

Consider,  now,  any  curve.  Divide  the  interval  from  XQ  to  xn  into  n 
parts  and  erect  the  ordinates  (Fig.  1052).  Through  A0,  AI,  A*  .  .  .  t 
draw  short  horizontal  lines.  Cut  the  arc  AoAi  by  a  vertical  line  making 


FIG.  IDS/. 

the  small  areas  bounded  by  this  vertical,  the  arc,  and  the  horizontals 
through  A0  and  A\,  equal.  Proceed  similarly  for  the  succeeding  arcs. 
Then  construct  the  integral  curve  of  the  stepped  line  by  the  method 
explained  above.  Choose  a  point  5  at  a  convenient  distance  a  to  the  left 
of  0  and  join  S  with  the  points  Co,  Ci,  C2,  .  .  .  ,  in  which  the  extended 


FIG. 


horizontals  cut  the  y-axis.  Then,  starting  at  B^  draw  a  line  through  J30 
parallel  to  SCo  until  it  cuts  the  first  vertical;  through  this  point  draw  a 
line  parallel  to  SCi  until  it  cuts  the  second  vertical,  etc.  The  points 
where  the  resulting  broken  line  cuts  the  ordinates  at  A0,  A\,  AZ,  .  .  .  , 
i.e.,  the  points  B0,  B\,  B^,  .  .  .  ,  are  points  on  the  required  integral  curve; 
for  at  each  of  the  points  A0,  A  i,  A*,  .  .  .  ,  the  area  under  the  curve  from 


242        APPROXIMATE   INTEGRATION   AND    DIFFERENTIATION         CHAP.  IX 

AO  to  that  point  is  equal  to  the  area  under  the  stepped  line;  so  that  a 
smooth  curve  through  the  points  .Bo,  Bi,  B%,  .  .  .  will  be  the  required 
integral  curve. 

Since  y'  =  -   I  y  dx,  therefore,  -~-  =  -  y,  so  that   the  slope   of    the 
a  J  J  dx       or 

integral  curve  at  any  point  is  proportional  to  the  ordinate  of  the  deriva- 
tive curve  at  the  corresponding  point.  Furthermore,  by  the  construc- 
tion, the  slopes  of  the  oblique  lines  through  B0,  B\,  Bi,  .  .  .  are  propor- 
tional to  the  ordinates  yQ,  yi,  y2,  .  .  .  ,  so  that  these  oblique  lines  are 
tangent  lines  to  the  required  integral  curve  at  these  points.  We  can 
thus  get  a  more  accurate  construction  of  the  integral  curve  by  drawing 
the  curve  through  the  points  B0,  Bi,  B2,  .  .  .  ,  tangent  to  the  oblique 
lines  through  these  points. 

The  polar  distance  SO  =  a  is  constructed  with  the  same  scale  unit  as 
the  abscissa  x,  and  the  ordinate  y'  is  measured  with  the  same  scale  unit 
as  the  ordinate  y. 

Example.  Determination  of  the  mean  spherical  candle-power  of  a  mazda 
lamp.  —  In  testing  a  lamp  for  the  m.  s.  c.  p.,  the  intensity  of  illumination 
is  measured  every  15°  by  means  of  a  rotating  lamp  and  a  photometer. 
The  following  table  gives  such  measurements  for  a  particular  case: 


Angle  0° 


c-p 


11-55 


15 
13.0 


30 
154 


45 


22.4 


60  I    75    I    90 
31.0  |  38.8  |  42.7 


105 


43-9 


45-2 


135 


32.0 


150 


21.8 


165 


9.1 


1 80 


According  to  the  well-known  Rousseau  diagram,  a  semicircle  is  drawn 
(Fig.  105/1)  and  divided  into  15°  sections,  and  perpendiculars  are  dropped 
from  the  points  of  division  to  the  diameter,  x0,  x\,  x2,  .  .  .  ,  #12-  Upon 
these  perpendiculars  the  values  of  c-p  are  laid  off  as  ordinates.  The  area 
under  the  curve  A§A\A<i  .  .  .An  determined  by  these  ordinates  divided 
by  the  length  of  the  base  is  the  m.  s.  c.  p.  of  the  lamp,  and  this  value 
multiplied  by  4  TT  will  give  the  flux  in  lumens. 

To  measure  the  required  area  we  have  constructed  the  integral  curve 
(Fig.  105/0  by  the  method  described  above.  We  chose  7  in.  for  the 
length  of  the  diameter  of  the  circle  and  I  in.  =  loc-p  in  laying  off  the  ordi- 
nates. The  y-axis  or  axis  to  which  the  horizontals  are  extended  is  drawn 
5  in.  to  the  right  of  the  point  A0,  so  that  the  polar  distance  is  A00  =  a 
=  Sin. 

The  area  under  the  curve  AoAiA2  .  .  .  A  12  is  measured  by  the  ordi- 

nate XnBit  =  4.66.     Since  /  =  -  X  area,   therefore  area  =  a  X  /  =  5 
X  4.66  =  23.3  sq.  in.     Since  I  in.  on  the  scale  of  ordinates  represents 


10  c-p  and  the  base  of  the  diagram  is  7  in.,  the  m.  s.  c.  p.  =  -=  — 

=  33-3  C~P-     The  straight  line  A^B^  will  cut  the  y-axis  in  a  point  D  such 
that  OD  read  on  the  c-p  scale  will  also  give  the  m.  s.  c.  p.,  for 


AKT.  105 


GRAPHICAL  INTEGRATION 


243 


or 


A00      A0xu 
We  measure  OD  =  33.0  c-p. 


base  base 


50 


FIG.  105/1. 


E.  L.  Ctor& 


Having  drawn   the   integral  curve  we  may  immediately  find   the 
m.  s.  c.  p.  of  any  portion  of  the  lamp  between  two  sections.     Thus,  for  15° 


244       APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION        CHAP.  IX 


on  each  side  of  the  vertical,  the  m.  s.  c.  p.  is  found  by  drawing  A0E  parallel 
to  B&Bi  and  reading  OE  =  42.0  c-p  on  the  candle-power  scale,  since 


OE 
a 


or     OE  = 


area  under  .4^47 
base 


m.  s.  c.  p. 


.152 


Similarly  the  m.  s.  c.  p.  of  the  section  above  a  horizontal  plane 
through  the  lamp  is  measured  by  OF  =  37.0  c-p,  and  the  m.  s.  c.  p.  of 
the  section  below  a  horizontal  plane  through  the  lamp  is  measured  by 
OG  =  29.5  c-p. 

106.   Graphical  differentiation.  —  If  the  integral  curve  /  =/(*)  is 

dy' 

given  we  may  construct  the  derivative  curve  y  =  -~-  by  using  the  prin- 
ciple that  the  ordinate  of  the  derivative  curve 
at  any  point  P  (x,  y)  (Fig.  1060)  is  equal  to 
the  slope  of  the  integral  curve  or  of  the 
tangent  line  P'T  at  the  corresponding  point 
P' (*,/). 

The  practical  construction  of  the  deriva- 
tive curve  consists  of  the  following  steps: 
(i)  Divide  the  interval  from  x0  to  xn  (Fig. 
1066)  into  n  parts  and  erect  the  ordinates 
yo,  y\  ,  y^,  •  •  •  ,  yn'-     (2)  Construct  the 
,  Bn  and  measure  their  slopes.     (3)  At  x0, 
y0,  XiAi  =  yi,  .  .  .  ,  xnAn  =  yn,  where 


P'(X,V) 


T 
FIG.  io6a. 


tangents  at  Bo,  B\,  B%,  , 

Xi,  .  .  .  ,  xn  erect  ordinates  XoA 


FIG.  io6&. 


the  y's  are  proportional  to  the  corresponding  slopes,  and  draw  a  smooth 
curve  through  the  points  A0,  A\,  A2,  .  .  .  ,  An.  This  curve  will  ap- 
proximate the  required  derivative  curve. 


ART.  106 


GRAPHICAL   DIFFERENTIATION 


245 


Example.  The  following  table  gives  the  pressure  p  in  pounds  per 
sq.  in.  of  saturated  steam  at  temperature  6°  F.  Construct  the  curve 
showing  the  rate  of  change  of  pressure  with  respect  to  the  temperature, 
dp/dB. 


e 

P 

Ap 

A0 

A,/A, 

302.7 
307-4 
311.8 
316.0 
320.0 
323-9 
327.6 

70 

75 
80 

85 
90 

95 

100 

5 
5 
5 
5 
5 
5 
5 

4-7 
4-4 
4.2 
4.0 
3-9 
3-7 
3-5 

.06 
•14 
•19 
•25 
.28 
•35 
•43 

334-5 
•  337-8 

no 
"5 

5 
5 

3-4 
3-3 

•47 
•52 

In  the  above  table  we  have  approximated  dp/dd  by  A£/A0,  i.e., 'we 
have  replaced  the  (6,  p)  curve  by  a  series  of  chords,  and  the  slopes  of  the 
tangents  by  the  slopes  of  these  chords.  We  then  plotted  (6,  A£/A0)  and 
joined  the  points  by  a  smooth  curve  (Fig.  io6c). 


. 

x 

1  aA 

JfO 

, 

/ 

1.0  (t 

•^ 

^^" 

]00 

tr\\ 

U 

^/^ 

7.4W 

uv 

( 

M 

£- 

.  —  • 

/ 

^ 

*^ 

^ 

-—  •— 

X 

f.2ff 

• 

x 

X 

t?r 

tjl 

SU 

^J 

? 

1.00 

70 

x 

^ 

00 

31 

0 

F 

5. 

( 

[G. 

70 

0 
1  06 

c. 

56 

0 

34 

0 

It  is  evident  that  the  difficulty  in  the  construction  of  the  derivative 
curve  lies  in  the  construction  of  the  tangent  line  to  the  integral  curve. 
The  direction  of  the  tangent  line  at  any  point  is  not  very  well  defined  by 
the  curve.  As  a  rule  it  is  better  to  draw  a  tangent  of  a  given  direction 
and  then  mark  its  point  of  contact  than  to  mark  a  point  of  contact  and 
then  try  to  draw  the  tangent  at  this  point.  A  strip  of  celluloid  on  the 
under  side  of  which  are  2  black  dots  about  2  m.m.  apart  may  be  moved 
over  the  paper  so  that  the  two  dots  coincide  with  points  on  the  integral 
curve  and  so  that  the  secant  line  which  they  determine  is  practically 
identical  with  the  tangent  line.  If  the  arc  AB  (Fig.  io6d)  is  approxi- 


246       APPROXIMATE   INTEGRATION   AND   DIFFERENTIATION         CHAP.  IX 

mately  the  arc  of  a  parabola,  we  have  a  more  accurate  construction  of  the 
tangent;  the  line  joining  the  middle  points  M  and  M'  of  two  parallel 
chords  AB  and  A'B'  intersects  the  curve  in  P,  the  point  of  contact, 
and  the  tangent  PT  is  parallel  to  the  chord  AB. 

We  may  also  draw  the  derivative 
curve  by  purely  graphical  methods. 
The  process  is  the  reverse  of  the 
process  described  for  constructing 
the  integral  curve  (Art.  105).  Let 
Bo,  Bi,  B2,  .  .  .  be  the  points  of 
contact  of  tangent  lines  to  the  in- 
tegral curve  (Fig.  105^).  Choose  a 

fixed  point  5  at  a  convenient  dis- 

'  tance  a  to  the  left  of  the,;y-axis  and 

draw  the  lines  SCo,  SCi,  SC2,  .  .  .  , 

parallel  respectively  to  the  tangent  lines  at  B0,  Bi}  Bi,  .  .  .  .  Project 
the  points  C0,  C\,  Ci,  .  .  .  ,  horizontally  on  the  ordinates  at  B0,  Bi, 
Bz,  •  •  •  ,  cutting  these  ordinates  in  A0,  A\,  A%,  ....  The  points 
Ao,  Ai,  A-i,  .  .  .  ,  arc  then  points  on  the  required  derivative  curve, 
since  B0A0  -*-  a  =  slope  of  SC0  =  slope  of  tangent  at  B0,  etc.  We  may 
now  join  the  points  A0,  AI,  AZ,  ...  by  a  smooth  curve,  or  we  may  get 
greater  accuracy  by  using  the  stepped  line  of  horizontals  and  verticals. 
Thus,  we  draw  the  horizontals  through  the  points  A0,  AI,  A2,  .  .  .  ,  and 
the  verticals  through  the  points  of  intersection  of  consecutive  tangents 
to  the  integral  curve.  The  arcs  AQ/!.},  AiAz,  .  •  •  ,  are  now  drawn  so 
that  the  areas  bounded  by  each  arc,  the  horizontals,  and  the  vertical, 
are  equal. 

107.  Mechanical  integration.*  The  planimeter.  —  This  is  an  in- 
strument for  measuring  areas.  Consider  a  line  PQ  of  fixed  length  / 
moving  in  any  manner  whatever  in  the  plane  of  the  paper.  The  motion 
of  the  line  at  any  instant  may  be  thought  of  as  a  motion  of  translation 
combined  with  a  motion  of  rotation.  Suppose  the  line  PQ  sweeps  out 
the  elementary  area  PQQ'P'  =  dS  (Fig.  loya).  This  may  be  broken  up 
into  a  motion  of  translation  of  PQ  to  P"Q'  and  a  motion  of  rotation 
from  P"Q'  to  P'Q'.  If  dn  is  the  perpendicular  distance  between  the 
parallel  positions  PQ  and  P'Q'  and  d<j>  is  the  angle  between  P"Q'  and 
P'Q',  then 

dS=  Idn  +  U2<f0. 

*  For  descriptions  and  discussions  of  various  mechanical  integrators  see:  Abdank- 
Abakanowicz,  Les  Integraphes  (Paris,  Oauthier-Villars) ;  Henrici,  Report  on  Planimeters 
(Brit.  Assoc.  Ann.  "Rep.,  1894,  p.  496);  Shaw,  Mechanical  Integrators  (Proc.  Inst.  Civ, 
Engs.,  1885,  p.  75);  Instruments  and  Methods  of  Calculation  (London,  G.  Bell  &  Sons); 
Dyck's  Catalogue;  Morin's  Les  Appareils  d' Integration. 


ART.  107         MECHANICAL  INTEGRATION.     THE  PLANIMETER 


247 


Now  if  PQ  carries  a  rolling  wheel  W,  called  the  integrating  wheel,  whose 
axis  is  parallel  to  PQ  (Fig.  1076),  then,  while  PQ  moves  to  the  parallel 
position  P"Qf,  any  point  on  the  circumference  of  this  wheel  receives  a 
displacement  dn,  and  while  P"Qr  rotates  to  the  position  P'Q',  this  point 
receives  a  displacement  a  d$,  where  a  is  the  distance  from  Q  to  the  plane 


W 


FIG.  1070. 


FIG.  1076. 


of  the  wheel.     So  that,  as  PQ  sweeps  out  the  elementary  area  dS,  any 
point  on  the  circumference  of  the  wheel  receives  a  displacement 

ds  =  dn  -\-  a  d<f>. 
Therefore,  dS  =  I  ds  -  al  d</>  + 

Hence  the  total  area  swept  out  by  PQ  is 


ids  -  alfd<t>  +  %l2  C 


FIG.  io7c. 

Now,  if  PQ  comes  back  to  its  original  position  without  turning  com- 
pletely around,  then  the  total  angle  of  rotation  /  d<f>  =  o,  so  that 

s-it, 

where  5  is  the  total  displacement  of  any  point  on  the  circumference  of  the 
integrating  wheel. 

But  if  PQ  comes  back  to  its  original  position  after  turning  completely 
around,  then 

S  =  Is  -  2iral  +  irP. 


248       APPROXIMATE   INTEGRATION  AND    DIFFERENTIATION         CHAP.  IX 

The  most  common  type  of  planimeter  is  the  Amsler  polar  planimeter  * 
(Fig.  loyc).  Here,  Fig.  107^,  by  means  of  a  guiding  arm  OQ,  called  the 
polar  arm,  one  end  Q  of  the  tracer  arm  PQ^s  constrained  to  move  in  a 
circle  while  the  other  end  P  is  guided  around  a  closed  curve  c-c-c-  .  .  . 
which  bounds  the  area  to  be  measured.  Then  the  area  Q'P'PP"Q"QQ' 
is  swept  out  twice  but  in  opposite  directions  and  the  corresponding  dis- 
placements of  the  integrating  wheel  cancel,  so  that  the  final  displacement 
gives  only  the  required  area  c-c-c-  ....  The  circumference  of  the 
wheel  is  graduated  so  that  one  revolution  corresponds  to]a  certain  definite 
number  of  square  units  of  area. 

P' 


FIG.  107^. 

The  ordinary  planimeter  used  for  measuring  indicator  diagrams  has 
/  =  4  in.  and  the  circumference  of  the  wheel  is  2.5  in.;  hence  one  revolu- 
tion corresponds  to  4  X  2.5  =  10  sq.  in.  The  wheel  is  graduated  into  10 
parts,  each  of  these  parts  again  into  10  parts,  and  a  vernier  scale  allows 
us  to  divide  each  of  the  smaller  divisions  into  10  parts,  so  that  the  area 
can  be  read  to  the  nearest  hundredth  of  a  sq.  in.  The  indicator  diagram 
on  p.  228  gives  a  planimeter  reading  of  2.55  sq.  in.,  which  agrees  with  the 
result  found  by  Simpson's  rule  with  15  ordinates. 

The  polar  planimeters  used  in  the  work  in  Naval  Architecture  usually 
have  a  tracer  arm  of  length  8  in.,  and  a  wheel  of  circumference  2.5  in.,  so 
that  one  revolution  corresponds  to  20  sq.  in.,  thus  giving  a  larger  range 
for  the  tracing  point.  If  the  area  to  be  measured  is  quite  large,  it  may  be 
split  up  into  parts  and  the  area  of  each  part  measured;  or  the  area  may 
be  re-drawn  on  a  smaller  scale  and  the  reading  of  the  wheel  multiplied 
by  the  area-scale  of  the  drawing. f 

*  This  instrument  was  first  put  on  the  market  by  Amsler  in  1854. 

f  If  PQ  (Fig.  107  d)  turns  completely  around,  the  required  area  is  5  +  T  (OQY. 


ART.  107 


MECHANICAL  INTEGRATION.      THE  PLANIMETER 


249 


If  very  accurate  results  are  required,  account  must  be  taken  of  several 
errors,  (i)  The  axis  of  the  integrating  wheel  may  not  be  parallel  to  the 
tracer  arm  PQ.  This  error  can  be  partly  eliminated  by  taking  the  mean 
of  two  readings,  one  with  the  pole  0  to  the  left  of  the  tracer  arm,  the  other 
with  the  pole  to  the  right*  (Fig.  loye). 
This  cannot  be  done  with  the  ordinary 
Amsler  planimeter  because  the  tracer 
arm  is  mounted  above  the  polar  arm, 
but  can  be  done  with  any  of  the  Coradi 
or  Ott  compensation  planimeters;  one 
of  these  instruments  is  illustrated  in 
Fig.  loj/.  (2)  The  integrating  wheel 
may  slip;  some  of  this  slipping  may  be 
due  to  the  irregularities  of  the  paper 
and  has  been  obviated  by  the  use  of 
disc  planimeters,  in  which  the  recording 
wheel  works  on  a  revolving  disc  instead  FlG 

of  on  the  surface  of  the  paper. 

Various  types  of  linear  planimeters  have  been  constructed.     These 
differ  from  the  polar  planimeters  in  that  one  end  of  the  tracer  arm  is 


FIG. 


constrained  to  move  in  a  straight  line  instead  of  in  a  circle.     Planimeters 
of  the  linear  type  form  part  of  the  integrators  described  in  Art.  108. 

Various  other  types  of  planimeters 
have  been  constructed,  which  do  not 
have  an  integrating  wheel.  One  of  the 
best  known  of  these  is  that  of  Prytz, 
also  known  as  the  hatchet  planim- 
eter. f  In  this  form  of  the  instrument 
(Fig.  ioyg)  the  end  Q  forms  a  knife- 
edge  so  that  Q  can  only  move  freely 
along  the  line  PQ.  When  P  traces  the 


FIG.  iojg. 

given  curve,  Q  will  describe  a  curve  such  that  PQ  is  always  tangent  to  it. 

*  For  a  proof  of  this  statement,  see  Instruments  and  Methods  of  Calculation,  p.  196. 
t  For  the  theory  of  this  instrument,  see  F.  W.  Hill,  Phil.  Mag.,  xxxviii,  1894,  p.  265. 


250       APPROXIMATE   INTEGRATION  AND   DIFFERENTIATION         CHAP.  IX 

Prytz  starts  the  instrument  with  the  point  P  approximately  at  the 
center  of  gravity  G  of  the  area  to  be  measured,  moves  P  along  the  radius 
vector  to  the  curve,  completely  around  the  curve,  and  back  along  the 
same  radius  vector  to  G.  The  required  area  is  then  given  approximately 
by  J20,  where  /  is  the  length  PQ  and  0  is  the  angle  between  the  initial 
and  final  positions  of  the  line  PQ. 

1 08.  Integrators.  —  The  Amsler  integrator  is  practically  an  extension 
of  the  linear  planimeter.  In  the  latter  instrument,  the  end  Q  of  the  tracer 
arm  PQ  of  constant  length  /,  is  constrained  to  move  in  a  straight  line  X'X, 
while  the  tracing  point  P  describes  a  circuit  of  the  curve.  If  the  axis  of 
the  integrating  wheel  attached  to  PQ  makes  a  variable  angle  ma  with 
X'X  (Fig.  io8a)  at  each  instant,  the  point  P  will  have  for  ordinate 

ym  =  I  sin  ma,  and  the  area  described  by  P  will  be   I  /  sin  ma  dx.     On 

the  other  hand,  the  area  described  by  P  is  equal  to  /  times  the  displace- 
ment of  any  point  on  the  circumference  of  the  integrating  wheel;  hence 

/  sin  ma  dx  is  equal  to  the  displacement  of  a  point  on  the  circumference  of 
an  integrating  wheel  whose  axis  makes  an  angle  ma  with  X'X. 


FIG.  io8a. 


FIG.  1086. 


Now,  given  a  curve  c-c-c-  .  .  .   (Fig.  1086), 

Area  =    I  y  dx  =    /  /  sin  a  dx  =  I   I  sin  a  dx. 

Moment  of  area       i    f  ,  ,        if,,..  PC,  \  j 

v/  v       =  -   I  -y2  dx  =  -   I  I2  sin2  a  dx  =  -    I  (l  —  cos  2  a)  dx 
about  X'X  2j^  2J  4  J 

I2  C          I2   C 

=  -  I  dx  --    I  sin  (90°  —  2a)dx 
4J  4  J 

-   I  sin  (90°  —  2  a)  dx,     since       I  dx  =  o, 

the  arm  PQ  returning  to  its  original  position  when  P  makes  a  complete 
circuit  of  the  curve. 


Moment  of  inertia  _  i_ 
of  area  about  X'X  ~  -\ 


- 

3  1'  \4 


i3  r .      .       i3  r .        , 

=  -  I  sm  a  dx I  sin  3  a  dx. 


ART.  108 


INTEGRATORS 


251 


Now 


,    I  sin  a  dx,     I  sin  (90°  —  2  a)  dx,  and     I  sin  3  a  dx,  and  hence 

the  area,  moment,  and  moment  of  inertia  can  be  measured  by  three  in- 
tegrating wheels  whose  axes  at  any  instant  make  angles  a,  90°  —  2  a, 
and  3  a,  respectively,  with  X'X. 

The  Amsler  $-wheel  integrator  (Fig.  io8c)  consists  of  an  arm  PQ  and 
3  integrating  wheels  A ,  M,  and  /.  The  instrument  is  guided  by  a  carriage 
which  rolls  in  a  straight  groove  in  a  steel  bar;  this  bar  may  be  set  at  a 
proper  distance  from  the  hinge  of  the  tracer  arm  by  the  aid  of  trams.  The 


FIG.  io8c. 


line  X'X,  which  passes  through  the  points  of  the  trams  and  under  the 
hinge,  is  the  axis  about  which  the  moment  and  moment  of  inertia  are 
measured.  The  radius  of  the  disk  containing  the  Af-wheel  is  one-half 
the  radius,  and  the  radius  of  the  disk  containing  the  /-wheel  is  one-third 
the  radius  of  the  circular  disk  D  to  which  they  are  geared.  Therefore, 
the  axis  of  the  M-wheel  turns  through  twice,  and  the  axis  of  the  /-wheel 
turns  through  three  times  the  angle  through  which  the  tracer  arm  PQ  or 
the  axis  of  the  A  -wheel  swings  from  the  axis  X'X. 

The  integrating  wheels  are  set  so  that  in  the  initial  position,  i.e.,  when 
P  lies  on  X'X,  the  axes  of  the  A-  and  /-wheels  are  parallel  to  X'X  while 
the  axis  of  the  M-wheel  is  perpendicular  to  X'X.  Then,  when  the  tracer 
arm  PQ  makes  an  angle  a  with  X'X,  the  axes  of  the  A-,  M-,  and  /-wheels 
make  angles  a,  90°  —  2  a,  and  3  a,  respectively,  with  X'X.  Further- 
more, the  graduations  of  the  M-wheel  are  marked  so  that  these  gradua- 
tions move  backward  while  the  graduations  on  the  other  wheels  move 


252       APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION        CHAP.  IX 

forward.  Hence,  when  P  has  completed  the  circuit,  and  if  a,  m,  and  i  are 
the  displacements  of  points  on  the  circumferences  of  the  A-,  M-,  and  I- 
wheels,  respectively,  we  have 

I2                                            P          /* 
Area  —  la;     Moment  =  -m;     Moment  of  Inertia  =  -a i. 

The  wheels  are  graduated  from  i  to  10  so  that  a  reading  of  5,  for 
example,  means  5/10  of  a  revolution.  The  constants  by  which  these 
readings  are  multiplied  depend  upon  the  length  of  the  tracing  arm  and 
the  circumferences  of  the  integrating  wheels.  In  the  ordinary  instru- 
ment, /  =  8  in.  and  the  circumferences  of  the  A-,  M-,  and  /-wheels  are 

CA  =  2.5  in.,         CM  =  2.5  in.,         d  =  2.34375  in. 
Thus,  to  find  the 
area,  a  must  be  multiplied  by'  8  X  2.5  =  20; 

moment,  m      "      "         "  "  -  X  2.5  =  40; 

4 

83 

moment  of  inertia,  a      "      "  "  —  X  2.5  =  320, 

4 

83 
and  *  "  —  X  2.34375  =  100. 

Finally,  if  a\,  az,  m\,  mz,  and  i\,  iz  are  the  initial  and  final  readings  of 
the  A-,  M-,  and  /-wheels,  we  have 

Area  =  20  (az  —  a\);     Moment  =  40  (w2  —  m\) ; 

Moment  of  Inertia—  320  (az  —  a\)  —  100  (iz  —  ii). 

109.  The  integraph.  —  This  is  a  machine  which  draws  the  integral 
curve,  y'  =  I  f(x)  dx,  of  the  curve  y  =  f(x} .  The  most  familiar  type  of 

such  machines  is  the  one  invented  by  Abdank-Abakanowicz  in  1878. 
The  theory  of  its  construction  is  very  simple.  A  diagram  of  the  machine 
is  given  in  Fig.  1090.  The  machine  is  set  to  travel  along  the  base  line  of 
the  curve  to  be  integrated,  and  two  non-slipping  wheels,  W,  ensure  that 
the  motion  continues  along  this  axis.  The  scale-bar  slides  along  the  main 
frame  as  the  tracing  point  P,  at  the  end  of  the  bar,  describes  the  curve 
y  =  f(x)  to  be  integrated.  The  radial-bar  turns  about  the  point  Q  which 
is  at  a  constant  distance  a  from  the  main  frame.  The  motion  of  the  re- 
cording pen  at  P\  is  always  parallel  to  the  plane  of  a  small,  sharp-edged, 
non-slipping  wheel  w,  and  by  means  of  the  parallel  frame- work  ABCD, 
the  plane  of  the  wheel  w  is  maintained  parallel  to  the  radial  bar  [since  w 
is  set  perpendicular  to  AB  which  is  parallel  and  equal  to  CD  throughout 
the  motion,  and  the  radial  bar  is  set  perpendicular  to  CD].  As  the  point 
P  describes  the  curve  y  =  /(*),  the  angle  6  between  the  radial-bar  and  the 


ART.  109 


THE   INTEGRAPH 


253 


axis,  and  consequently  the  angle  6  between  the  plane  of  the  wheel  and  the 
axis,  are  constantly  changing,  and  the  recording  pen  at  PI  draws  a  curve 
with  ordinate  y'  such  that  its  slope 


dx 


and  therefore, 


y'  =  1   Cf  (x)  dx=^X  area  ORP, 


so  that  the  curve  drawn  by  PI  is  the  integral  curve  of  the  curve  traced 
by  P. 


W 


FIG.  1090. 


If  we  now  set  the  machine  so  that  the  point  P  traces  the  integral  curve, 
then  the  recording  pen  PI  will  draw  its  integral  curve 


We  may  thus  draw  the  successive  integral  curves  y',  y",  y'"  .....  Fig. 
1096  gives  the  integral  curves  connected  with  the  curve  of  loads  of  the 
shaft  of  a  Westinghouse-Rateau  Turbine.  The  curve  of  loads  is  repre- 
sented by  the  broken  line  in  the  figure.  By  successive  integration  we  get 
the  shear  curve,  the  bending  moment  curve,  the  slope  curve,  and  the 
deflection  curve.  The  distance  marked  "offset"  is  the  distance  OOi  in 
Fig. 


(254) 


ART.  no 


MECHANICAL  DIFFERENTIATION 


255 


no.   Mechanical  differentiation.    The  Differentiator.  —  This  is  a 

dy 

machine  which  draws  the  derivative  curve  y'  =  -/-  of  the  curve  y  =  f(x). 

CLx 

Since  the  ordinate  of  the  derivative  curve  is  equal  to  the  slope  of  the  in- 
tegral curve,  it  is  necessary  to  construct  the  tangent  lines  at  a  series  of 
points  of  the  integral  curve.  We  have  already  mentioned  (Art.  106)  the 
use  of  a  strip  of  celluloid  with  two  black  dots  on  its  under  side  to  deter- 


FIG.  no. 

mine  the  direction  of  the  tangent.  This  scheme  is  used  in  a  differentiating 
machine  constructed  by  J.  E.  Murray.*  In  a  differentiating  machine 
recently  constructed  by  A.  Elmendorf,f  a  silver  mirror  is  used  for  de- 
termining the  tangent.  The  mirror  is  placed  across  the  curve  so  that 
the  curve  and  its  image  form  a  continuous  unbroken  line,  for  then  the 
surface  of  the  mirror  will  be  exactly  normal  to  the  curve,  and  a  perpen- 
dicular to  this  at  the  point  of  intersection  of  the  mirror  and  the  curve  will 
give  the  direction  of  the  tangent  line.  If  the  surface  of  the  mirror  de- 

*  Proc.  Roy.  Soc.  of  Edinburgh,  May,  1904. 

f  Scientific  American  Supplement,  Feb.  12,  1916.] 


256       APPROXIMATE  INTEGRATION  AND  DIFFERENTIATION        CHAP.  IX 

viates  even  slightly  from  the  normal,  a  break  will  occur  at  the  point  where 
the  image  and  curve  join.  It  is  claimed  that  with  a  little  practice  a  re- 
markable degree  of  accuracy  can  be  obtained  in  setting  the  mirror. 

Fig.  no  gives  a  diagram  illustrating  the  working  of  this  machine. 
The  tracing  point  P  follows  the  curve  y  =  f(x)  so  that  the  curve  and  its 
image  in  the  mirror  MP  form  a  continuous  unbroken  line;  then  the  arm 
P  T,  which  is  set  perpendicular  to  the  mirror,  will  take  the  direction  of  the 
tangent  line  to  the  curve.  The  link  PR,  of  fixed  length  a,  is  free  to  move 
horizontally  in  the  slot  X'X'  of  the  carriage  C.  The  vertical  bar  SU 
passes  through  R  and  is  constrained  to  move  horizontally  by  heavy  rollers. 
The  point  Q  slides  out  along  the  tangent  bar  PT  and  also  vertically  in  the 
bar  SU,  carrying  with  it  the  bar  QPr.  If  we  choose  for  the  #-axis  a 
line  XX  whose  distance  from  X'X'  is  equal  to  QP't  then  the  point  P' 
will  draw  a  curve  whose  ordinate  is  equal  to  y'  =  RQ.  But  RQ/a  is 

the  slope  of  the  tangent  PT,  hence,  y'  =  a  X  -f^-,  and  the  curve  drawn 

by  P'  is  the  derivative  curve  of  the  curve  traced  by  P. 

The  machine  is  especially  useful  for  differentiating  deflection-time 
curves  to  obtain  velocity-time  curves,  and  by  a  second  differentiation, 
acceleration-time  curves.  It  is  also  helpful  in  solving  many  other 
problems. 


EXERCISES. 

Apply  the  approximate  rules  of  integration  (Art.  101)  to  the  following  examples: 

/*1-°  dx 

1.  Evaluate  J      —  ,  when  h  =  o.i,  and  when  h  =  0.05,  and  compare  the  results  with 

the  values  obtained  by  integration. 

2.  Evaluate   \     sin  x  dx,  when  h  =  ^  ,  and  when  h=  —  ,  and  compare  the  results 

•/o  6  12 

with  the  values  obtained  by  integration. 

3.  The  arc  of  a  quadrant  of  an  ellipse  whose  eccentricity  is  0.5   is  given  by 
•*• 

pi     .  _ 
J     vi  —  0.25  sin2<£  d<l>.     Evaluate  the  integral  when  h  =  9°. 

rdx 
,.  =.  ,  when  h  =  0.5. 

-    v  x3  -  x  +  i 

5.  The  semi-ordinates  in  ft.  of  the  deck  plan  of  a  ship  are 

3,       16.6,       25.5,       28.6,      29.8,      30,      29.8,      29.5,      28.5,      24.2,      6.8; 
these  measurements  are  28  ft.  apart.     Find  the  area  of  the  deck. 

6.  Given  the  following  data  for  superheated  steam 

io  ^ 
13 
Find  the  work  done. 


V 

2 

4 

6        1        8 

P 

105 

42.7 

25-3     I     16.7 

EXERCISES 


257 


7.  The  length  of  an  indicator  diagram  is  3.6  in.     The  widths  of  the  diagram,  0.3  in. 
apart,  are 

o,    0.40,    0.52,    0.63,    0.72,    0.93,    0.99,     i. oo,     i. oo,     i. oo,     i. oo,    0.97,    o. 
Find  the  mean  effective  pressure. 

8.  The  length  of  an  indicator  diagram  is  3.2  in.     The  widths  of  the  diagram,  0.2  in. 
apart,  are 

i. oo,         1.68,         1.62,         i. oo,         0.64,         0.48,         0.36,         0.26,         o. 
Find  the  mean  effective  pressure. 

9.  The  speed  of  a  car  is  v  miles  per  hour  at  a  time  t  seconds  from  rest; 


1 

o 

5        1       10 

15 

20 

25 

30 

V 

0 

3-7            7-5 

10.9 

13-0 

13-7 

H 

Find  the  distance  traversed  in  30  seconds. 


10.   5  is  the  specific  heat  of  a  body  at  temperature  6°  C. 
o  2          |          4  6 


1.00664       1-00543    I     1-00435         1-00331     |     1.00233    |     1.00149         1.00078 
Find  the  total  heat  required  to  raise  the  temperature  of  a  gram  of  water  from  o°  C.  to 

12°  C.  (total  heat  =    f^sde). 

•/BI 

11.  The  areas  in  sq.  ft.  of  the  sections  of  a  ship  above  the  keel  and  two  feet  apart  are 

2690,         3635,         4320,         4900,         5400. 
Find  the  total  displacement  in  tons. 

12.  A  reservoir  is  in  the  form  of  a  volume  of  revolution  and  D  is  the  diameter  in  ft. 
at  a  depth  of  p  feet  beneath  the  surface  of  the  water. 

p  o  16      |      32  48      |      64 80      I     96 


D  no          105     I     100  86      I      66  48      J     27 

Find  the  number  of  gallons  of  water  the  reservoir  holds  when  full. 

13.  A  plane  board  is  immersed  vertically  in  water.     The  widths  of  the  board  in  ft. 
parallel  to  the  surface  of  the  water  and  at  depths  £  ft-  apart  are 

4.0,         3.6,         3.0,         1.7,         1.3,         i.o,         0.8,         0.6,         o.i. 

Find  the  pressure  on  the  board  and  the  depth  of  the  center  of  pressure  when  the  surface 
of  the  water  is  level  with  the  top  of  the  board. 

14.  The  half-ordinates  in  ft.  of  the  mid-ship  section  of  a  vessel  at  intervals  2  ft. 
apart  are 

12.2,       12.5,       12.6,       12.7,       12.7,       12.5,       12. 1,       II.5,       IO.I,      6.5,      0.2. 

Find  the  position  of  the  center  of  gravity  of  the  section. 

15.  The  shape  of  a  quarter-section  of  a  hollow  pillar  is  given  by  the  following  table. 
The  axes  of  x  and  y  are  the  shortest  and  longest  diameters. 


x'm. 

o 

0.25 

0.50 

075 
5.83 

1.  00 

5-64 

ISO 

5-48 

ITS 

5-22 

2.00 

2.25 

2.50_ 

4-35 

^75 
3-88 

3-00 
3-25 

3-25 

out- 
side 
y\  in. 
in- 
side 
y*  in. 

6 

5-95 

5-90 

5T6 

4-99 

4.68 

2-34 

5 

4.90 

4.78 

4-65 

4-45 

4.22 

3-8o 

3-40 

2-77 

2.08 

o 

Find  the  moments  of  inertia  of  the  section  about  the  x-  and  y-  axes. 


258       APPROXIMATE   INTEGRATION  AND   DIFFERENTIATION        CHAP.  IX 


16.  Apply  the  formulas  for  numerical  differentiation  (p.  235)  to  table  (2)  y  =  x3  on 
p.  211,  and  find  ~  and  -3-^  when  x  =  5.31  and  *  =  5.33.     Check  the  results  by  actual 
differentiation. 

17.  Apply  the  formulas  for  numerical  differentiation  (p.  235)  to  table  (8)  y  =  log  sin  x 
on  p.  212,  and  find  -3-  and  -j— -  when  x  =  i°  20'  and  x  =  i°  24'.     Check  the  results  by 
actual  differentiation. 

18.  In  the  following  table,  5  is  the  distance  in  ft.  which  the  projectile  of  a  gun  travels 
along  the  bore  in  /  sec. 


0.0360  |  0.0490!  0.0598 


0.0695 


0.0785 


0.0871 


0-0953  I  0.1032  |  0.1109  I  0.1184 


ds         fdt  d?s          dH  //dt\3    , 

Find  the  velocity  v  =  -r  =  i  1  -7- ,  and  the  acceleration  a  =  -j^  =  ~  j~i  /  I  ;r  )  when  5  = 

5ft. 

19.  Use  the  data  given  in  Ex.  6  to  find  the  rate  of  change  of  the  pressure  with  re- 
spect to  the  volume,  dp/dv,  when  v  =  4  and  v  =  5. 

20.  Use  the  data  given  in  Ex.  9  to  find  the  acceleration,  a  =  -j-  ,  when  /  =  10  and 
t  =  12. 

21.  Find  the  minimum  value  of  the  polynomial  which  has  the  values 

4        I        6 
ii  27 

22.  The  following  table  gives  the  results  of  measurements  made  on  a  normal  in- 
duction curve  for  transformer  steel;  B  is  the  number  of  kilolines  per  sq.  cm.;  ju  is  the  per- 
meability. 

B 


625 


870 


1035 


1350 


1465 


1520  I  1480  |    1430 


TO 


1370 


II 


1280 


1130 


Find  the  maximum  permeability. 

23.  Construct  the  integral  curve  of  the  parabola  y  =  x  —  |  x-  as  x  varies  from  o  to  2. 

24.  Construct  the  integral  curve  of  the  sine  wave  y  =  2  sin  2  x  as  x  varies  from  o  to  IT. 

25.  The  following  table  gives  the  accelerations  a  of  a  body  sliding  down  an  inclined 
plane  for  various  distances  5  in  ft. 

s  \    o    |  loo      200       300    [    400        500     |     600         700         800     |     900     I    1000 


-I— I- 
a  [o.32o|( 


a  [0.32010.304  0.256    0.176  |  0.080    -0.016  | -0.080    -0.136-0.1761-0.2081-0.240 
Use  the  method  employed  in  the  illustrative  example  on  p.  239  for  drawing  the  integral 

curves  and  determining  the  velocity,  v  -  y  2  fa  ds,  and  the  time,  t  =  J  -  ds,  for  any 
distance,  if  v  =  o  and  /  =  o  when  s  =  o. 

26.   The  following  table  gives  the  accelerations  a  of  a  body  at  various  velocities  v  in 
ft.  per  sec. 


0.405 


0.360    |    0.283 


Draw  the  integral  curves  to  determine  the  time,  t 
for  any  velocity,  if  /  =  o  and  5  =  0  when  v  =  o. 


0.179        0.069    I    0.013 

/- dv,  and  the  distance,  s  =  \  -dv, 
a  J  a     ' 


EXERCISES 


259 


27.    In  the  following  table 


o 

I 

4 

6 

8 

"•5 

15 

38,000 

38,500 

38,500 

35,500 

27,500 

19,000 

15,700 

xx)  I    3850 

P  is  the  resultant  pressure  in  pounds  on  the  piston  of  a  steam  engine  at  distances  s  inches 
from  the  beginning  of  the  stroke.     Draw  the  integral  curve  to  find  the  work  done  as  the 

piston  moves  forward.     (  Work  =  J  P  ds.  J 

28.   A  car  weighs  10  tons.     It  is  drawn  by  a  pull  of  P  Ibs.;  /  is  the  time  in  seconds 
since  starting. 
t  o  2  5 |        8        |       10  13       |      16       I      19 


P     1020     980      882   I   720   I   702      650   I   713   I   722      805 

If  the  retarding  friction  is  constant  and  equal  to  410  Ibs.,  draw  the  integral  curve  to  find 
the  speed  of  the  car  at  any  time.     (  Momentum  =  J  (P  —  410)  dt.  J 

29.    In  the  following  table 
t     I  0.00490    0.00598  I  0.00695   0.0078510.00871   0.00953  1 0.01032   0.01109    0.01184 


869  987      |     1074         H42    I    H95        1242    |    1277         1309         1335 

v  is  the  velocity  of  projection  in  ft.  per  sec.  in  the  bore  of  a  gun  at  time  t  sec.  from  the 
beginning  of  the  explosion.  U  s  =  2  ft.  when  /  =  0.00490  sec.,  draw  the  integral  curve 
to  show  the  relation  between  the  distance  and  the  time. 

30.   A  beam  10  ft.  long  is  loaded  as  in  the  following  table,  where  w  is  the  weight  per 
unit  length  at  distances  x  ft.  from  the  free  end. 


2.5 


3-7    I    5-5 


4 

5 

6 

7 

8 

9 

IO 

7-7 

9-7 

11.2 

12.2 

n.8 

10.2 

7.2 

Draw  integral  curves  to  show  (i)  the  shearing  force,  s  =    \w  dx  and  (2)  the  bending 

moment,  M  =  (  s  dx. 

31.   The  following  table  gives  the  measurements  for  every  15°  of  the  intensity  of 
illumination  of  a  lamp. 


Angle  6° 


c-p 


_2_J5_ 
60  5!  88.0 


99-5 


45  I  60 

86.5 ;  50.0 


_75__90_ 


25.0 


28.0 


135 

20.0 


150 

165 

180 

15.0 

13-0 

12.5 

Apply  the  method  of  the  illustrative  example  on  p.  242  to  find  the  m.s.c.p.  for  various 
sections  of  the  lamp. 

32.    In  the  following  table 


IO 

20 

30 

156 

608 

1308 

60 

70 

80 

90 

100 

4076 

4942 

5676 

6236 

6588 

5  is  the  distance  in  ft.  traversed  by  a  body  weighing  2000  Ibs.  in  t  sec.  Draw  the  deriva- 
tive curves  to  show  the  velocity  and  acceleration  at  any  time.  Also  draw  the  curve 
showing  the  relation  between  the  kinetic  energy  and  the  force. 

33.   The  observed  temperature  6  in  degrees  Centigrade  of  a  vessel  of  cooling  water 
at  time  t  in  min.  from  the  beginning  of  observation  are  given  in  the  following  table: 

t    I 

0  I    < 
Draw  the  derivative  curve  to  show  the  rate  of  cooling  at  any  time. 


I 

2 

3 

5 

7 

10 

15 

20 

85-3 

79-5 

74-5 

67.0 

60.5 

53-5 

45-o 

39-5 

INDEX. 


Adiabatic  expansion  formula,  48 

chart  for,  33,  49 

Alignment  or  nomographic  charts  (see 
also  Charts,  alignment  or  nomo- 
graphic) 

fundamental  principle  of,  44 
with  curved  scales,  106 
with  four  or  more  parallel  scales,  55 
with    parallel    or    perpendicular    index 

lines,  87,  91,  97 

with  three  or  more  concurrent  scales,  104 
with  three  parallel  scales,  45 
with  two  intersecting  index  lines,  68 
with   two   or  more   intersecting   index 

lines,  76 

with  two  paraHel  scales  and  one  inter- 
secting scale,  65 

Approximate  differentiation,  224 
Approximate  integration,  224 
Area, 

by  approximate  integration  rules,  227 
by  planimeter,  246 

Armature  or  field  winding  formula,  90 
chart  for,  90 

Bazin  formula,  101 
chart  for,  102,  116 

Center  of  gravity,  by  approximate  inte- 
gration rules,  231 

Chart,  alignment  or  nomographic,  for 
adiabatic  expansion,  49 
armature  or  field  winding,  90 
Bazin  formula,  102,  116 
Chezy  formula  for  flow  of  water,  58 
D'Arcy's  formula  for  flow  of  steam,  81 
deflection  of  beams,  72,  73,  86 
discharge  of  gas  through  an  orifice,  89 
distributed  load  on  a  wooden  beam,  83 
focal  length  of  a  lens,  106 
Francis  formula  for  a  contracted  weir, 

109 
friction  loss  in  pipes,  94 


Chart,  Grasshoff's  formula,  51 

Hazen- Williams  formula,  60 

horsepower  of  belting,  54 

indicated    horsepower   of   a  steam  en- 
gine, 63 

Lame  formula  for  thick  hollow  cylin- 
ders, 92 

McMath  "run-off,"  formula,  49 

moment  of  inertia  of  cylinder,  100 

multiplication  and  division,  47 

prony  brake,  70 

resistance  of  riveted  steel  plate,  103 

solution  of  quadratic  and  cubic  equa- 
tions, 112 

specific    speed    of    turbine    and    water 
wheel,  75 

storm  water  run-off  formula,  108 

tension  in  belts,  54 

tension  on  bolts,  67 

twisting  moment  in  a  cylindrical  shaft, 
78 

volume  of  circular  cylinder,  49 

volume  of  sphere,  49 
Charts,  hexagonal,  40 
Chart  with  network  of  scales,  for 

adiabatic  expansion,  33 

chimney  draft,  38 

elastic  limit  of  rivet  steel,  34 

equations  in  three  variables,  28 

equations  in  two  variables,  20 

multiplication  and  division,  30,  31 

solution  of  cubic  equation,  36 

temperature  difference,  39 
Chezy  formula  for  flow  of  water,  56 

chart  for,  58 
Chimney  draft  formula,  37 

chart  for,  38 

Coefficients  in  trigonometric  series  evalu- 
ated, 

by  six-ordinate  scheme,  179 

by  twelve-ordinate  scheme,  181 

by  twenty-four-ordinate  scheme,  185 

for  even  and  odd  harmonics,  179 


INDEX 


Coefficients  in  trigonometric  series  evalu- 
ated, 

for  odd  harmonics  only,  1 86 

for  odd  harmonics  up  to  the  fifth,  187 

for  odd  harmonics  up  to  the  eleventh, 
189 

for  odd  harmonics  up  to  the  seventeenth, 
191 

graphically,  2OO 

mechanically,  203 

numerically,  179,  186,  192,  198 
Constants    in    empirical    formulas    deter- 
mined by 

method  of  averages,  124,  126 

method  of  least  squares,  124,  127 

method  of  selected  points,  124,  125 
Coordinate  paper, 

logarithmic,  22 

rectangular,  21 

semilogarithmic,  24 

D'Arcy's  formula  for  flow  of  steam,  79 

chart  for,  8 1 
Deflection  of  beams,  70,  71,  84 

chart  for,  72,  73,  86 
Differences,  210 
Differentiation,  approximate,  224 

graphical,  244 

mechanical,  255 

numerical,  234 
Differentiator,  255 
Discharge  of  gas  through  an  orifice,  89 

chart  for,  89 
Distributed  load  on  a  wooden  beam,  80 

chart  for,  83 
Durand's  rule,  226 

Elastic  limit  of  rivet  steel,  32 

chart  for,  34 
Empirical  formulas, 

determination  of  constants  in,  124,  125, 

173,  174 

for  non-periodic  curves,  120 
for  periodic  curves,  170 
involving  2  constants,  128 
involving  3  constants,  140 
involving  4  or  more  constants,  152 
Equations,  solutions  of  (see  Solutions  of 

algebraic  equations) 
Experimental  data,  120,  170 
Exponential  curves,   131,    142,    151,    153, 
156 


Focal  length  of  a  lens, 

chart  for,  35,  40,  106 

slide  rule  for,  15 
Fourier's  series,  170 
Francis  formula  for  a  contracted  weir,  no 

chart  for,  109 
Friction  loss  in  pipes,  94 

chart  for,  94 
Fundamental  of  trigonometric  series,  170 

Gauss's  interpolation  formula,  219 
Graphical  differentiation,  244 
Graphical  evaluation  of  coefficients,  200 
Graphical  integration,  237 
Graphical  interpolation,  209 
Grasshoff's  formula,  50 
chart  for,  51 

Harmonic  analyzers,  203 

Harmonics  of  trigonometric  series,  170 

Hazen-Williams  formula,  57 

chart  for,  60 
Hexagonal  charts,  40 
Horsepower  of  belting,  52 

chart  for,  54 
Hyperbola,  149 
Hyperbolic  curves,  128,  135,  137,  140 

Index  line,  44 

Indicated  horsepower  of  steam  engine,  6l 

chart  for,  63 
Integraph,  252 
Integration,  approximate,  224 

applications  of,  227 

by  Durand's  rule,  226 

by  rectangular  rule,  223 

by  Simpson's  rule,  226,  233 

by  trapezoidal  rule,  225 

by  Weddle's  rule,  233 

general  formula  for,  231 

graphical,  237 

mechanical,  246 
Integrators,  250 
Interpolation,  209 

Gauss's  formula  for,  219 

graphical,  209 

inverse,  219 

Lagrange's  formula  for,  218 

Newton's  formula  for,  214,  217 
Isopleth,  44 

Lagrange's  interpolation  formula,  218 


INDEX 


xiii 


Lame  formula  for  thick  hollow  cylinders, 

91 

chart  for,  92 

Least  Squares,  method  of,  124,  127 
Logarithmic  coordinate  paper,  22 
Logarithmic  curve,  151 
Logarithmic  scale,  » 

Maxima    and    minima    by    approximate 

differentiation  formulas,  236 
McMath  "run-off"  formula,  48 

chart  for,  49 
Mean  effective  pressure  by  approximate 

integration  rules,  228 
Mechanical  differentiation,  ,255 
Mechanical  integration,  246 
Moment,  by  integrator,  250 
Moment  of  inertia, 

by  approximate  integration  rules,  230 

by  integrator,  250 
Moment  of  inertia  of  cylinder,  99 

chart  for,  100 

Multiplication   and   division,   charts   for, 
30,  31,  41,  47 

Newton's  interpolation  formula,  214,  217 
Nomographic    or    alignment    charts    (see 

Alignment  or  nomographic  charts) 
Numerical  evaluation  of  coefficients,  179, 

186,  192,  198 

Numerical  differentiation,  234 
Numerical  integration,  224 
Numerical  interpolation,  215 

Parabola,  145 

Parabolic  curves,  128,  135,  140 

Periodic    phenomena,    representation    of, 

170 
Planimeter, 

Amsler  polar,  248 

compensation,  249 

linear,  249 

principle  of,  246 
Polynomial,  159 
Pressure  and  center  of  pressure,  by 

approximate  integration  rules,  231 
Prony  brake,  69 

chart  for,  70 

Rates  of  change,  by  approximate  differ- 
entiation formulas,  235 


Rectangular  coordinate  paper,  21 
Rectangular  rule,  225 
Resistance  of  riveted  steel  plate,  101 
chart  for,  103 

Scale, 

definition  of,  I 

equation  of,  2 

logarithmic,  2 

representation  of,  I 
Scale  modulus,  2 
Scales, 

network  of,  20 

perpendicular,  20 

sliding,  7 

stationary,  5 

Semilogarithmic  coordinate  paper,  24 
Simpson's  rule,  226,  233 
Slide  rule, 

circular,  16 

for  electrical  resistances,  15 

for  focal  length  of  lens,  15 

Lilly's  spiral,  1 8 

logarithmic,  9 

log-log,  13 

Sexton's  omnimetre,  17 

Thacher's  cylindrical,  18 
Solutions  of  algebraic  equations, 

by  means  of  parabola  and  circle,  26 

by  means  of  rectangular  chart,  35 

by  means  of  alignment  chart,  no 

by  method  of  inverse  interpolation,  221 

on  the  logarithmic  slide  rule,  1 1 
Specific  speed  of  turbine  and  water  wheel, 
73 

chart  for,  75 
Storm  water  run-off  formula,  107 

chart  for,  108 
Straight  line,  122,  125 

Tables,  construction  of,  213 
Temperature  difference,  37 

chart  for,  39 
Tension  in  belts,  52 

chart  for,  54 
Tension  on  bolts,  66 

chart  for,  67 
Trapezoidal  rule,  225 
Trigonometric  series,  170 

determination  of  constants  in,  173,  174 
Twisting  moment  in  a  cylindrical  shaft,  77 

chart  for,  78 


J1V  INDEX 

Velocity,  by  approximate  integration  rules,  Volume  of  sphere,  50 

229  chart  for,  49 

Volume,  by  approximate  integration  rules, 

229  Weddle's  rule,  233 

Volume  of  circular  cylinder,   48  Work,  by  approximate  integration  rules, 
chart  for,  49  228 


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Series  9482 


A     000  572  233     5 


